A lot of smart people believe that Sen. Elizabeth Warren of Massachusetts is the front-runner for the Democratic nomination in 2020. A recent New York magazine profile by Jonathan Chait claimed that she’s winning the invisible primary — building campaign networks, rolling out policy proposals, wooing insiders, intimidating rivals.

When she released a biographical video featuring her Oklahoman roots and answering Donald Trump’s “Pocahontas” jibe with a DNA test proving that she does have Native American ancestry, Jonathan Bernstein of Bloomberg View called the video the “latest in a series of solid moves Warren has been making” to persuade Democratic insiders that she can take on Trump.

I am not a Democratic insider or a campaign professional, so for all I know Warren may indeed be impressing her intended audience. And I can certainly think of worse people for the Democratic Party to nominate than a woman who has white-working-class roots as well as academic credentials, and a half-buried past as a heterodox intellectual (go read her book “The Two-Income Trap,” seriously).

But running for president in the age of Trump requires, above all, an ability to handle what John Heileman and Mark Halperin once called “the freak show” (back when it was considerably less freaky). It requires a deftness dealing with scandals and gaffes and accidental blunders, an ability to know when you have a wrestling move that justifies getting down in the mud and when you’re better off sitting on a top rail and acting superior to the pigs.

So far Warren’s main encounter with the freak show has involved her claim to Cherokee ancestry, which was an issue in her last Senate campaign, in 2012, before Trump started in with his nicknaming. And from her initial response to the story through the new DNA test “rebuttal” to the president, she has demonstrated a conspicuous lack of political common sense.

Here are the basic facts. There was plausible Warren family lore, as there is in many Oklahoman families, about a Cherokee ancestor, which included a memory of in-law bigotry against Warren’s mother for her supposed Cherokee and Delaware blood. At some point in Warren’s academic career, this lore became part of her official biography, so that she was listed as a “Native American” professor at the University of Pennsylvania and described as Harvard Law’s “first woman of color,” and she even contributed a family recipe to a Native American cookbook.

When this story first surfaced six years ago, I wrote that it was more embarrassing for the Ivy League than for Warren. The identification didn’t seem to have improved her academic career — something confirmed subsequently by documentation from her hirings — and her brief period as a supposed minority trailblazer was, I suggested, probably a whim of self-identification that she subsequently regretted, and was happy enough to let slip away. In which case it was the schools’ eagerness to turn a woman they had hired on the merits into an embodiment of an essentially phony diversity that was the real problem, not Warren’s attachment to her family tales.

But what Warren should have done when the story resurfaced, what she obviously should have done, was to simply express mild regret for letting her enthusiasm for family lore carry her away into identifying as someone who might possibly receive affirmative-action consideration, apologize to Cherokee groups for any offense, and literally never speak of the matter again. And if and when Donald Trump started up his Pocahontas jibes, she should have simply ignored him and talked about the many issues where he’s on the wrong side of public opinion.

In the event, she didn’t apologize, presumably because she thought that she did have a legitimate claim to minority status. And then she — or some too-clever strategist with her ear — set out to prove it with a DNA test, so that she could throw it back in Trump’s yawping face.

And the DNA is, indeed, solid evidence ... that Warren has at least one Indian ancestor between six and 10 generations back. Contrary to certain instant conservative critiques, this is legitimately interesting genealogy performed by a legitimate expert. It suggests that she may have more Native American blood, not less, than the average white American. And while a crude reading of the estimate suggests that she’s between 1/64th and 1/1,024th Cherokee (or some other mix of tribes), it’s also still possible that she’s somewhat more Indian than that.

So checkmate, right? Well, no, not at all. Because the whole issue with Warren’s ancestral claims is that she allowed them to be proclaimed as proof of an elite institution’s diversity, in an environment where that diversity can matter a great deal to one’s career prospects (even if, on the evidence, it didn’t matter much to Warren’s). And from the identitarian left and the anti-affirmative-action right, I defy you to find a single person invested in these debates who believes that someone who might be 1/64th or even 1/32nd Indian and whose Cherokee experience consists of old family stories should qualify as a first “minority” hire in anything.

The DNA test thus simultaneously gives Trump an obvious way to keep the story going on his terms — just pick the lowest end of the genetic estimate and make sport of a “Pocahontas” who’s only 1/1,000th Indian — while also annoying Indian groups and anyone on the left (including the actual minority candidates against whom Warren may run) invested in a vision of affirmative action as a righter of historic wrongs rather than just a means to elite self-congratulation.

Warren should not have taken the test; having taken it, she should not have publicized it; having publicized it, she should quietly fire anyone who urged this gambit and move on. And liberals generally should regard this whole thing as a cautionary tale. There is an obvious appetite on the activist left for a candidate or candidates willing to take on Trump on his own brawler’s terms. But if you come at him that way, you best not miss — as Michael Avenatti, the would-be Trump of the Resistance, has been missing repeatedly of late, with a Kavanaugh intervention that helped get the judge confirmed and a libel lawsuit that just got his own client ordered to pay Trump’s legal fees.

Now Warren has joined Avenatti in that loser’s gallery, for absolutely no good reason that I can see. Her self-own isn’t the biggest deal in the world, and her partisans are right that if mostly conservatives are agitated by the fiasco then it might not hurt her much. But a “front-runner” who’s polling at 8 percent needs to be showing reasons other candidates should be pre-emptively intimidated, and why the party should decide on her. And the fact that we’re talking, for no good reason, about a fragment of Warren’s DNA three weeks before an essential midterm is a sign that she shouldn’t exactly terrify her rivals yet.