In John 8, why did the Jews call themselves "children of Abraham" not "children of Jacob" or something else?

Considering that in the Jewish mindset at the time of Jesus, as we learn from Paul in Romans 9:6-13, the Jews thought that they were "the children of promise" descended from Isaac (not Ishmael) and "God's elect" descended from Jacob (thus Paul citing God's election of Jacob over Esau), **why then, in John 8, the Judeans refer to themselves as "children of Abraham" rather than the "children of Jacob" or something else?** I understand that Paul used Abraham as a model of faith, but faith was not the topic in John 8:30-59. Why did Jesus's interlocutors refer to themselves as "children of Abraham"? Did they implicitly acknowledge that the *Idumeans* (roughly [the leftover of the Edomites in the Hellenistic period](https://en.wikipedia.org/wiki/Edom#Hellenistic_period) , descendants of Esau) and the *inhabitants of the Nabataean Kingdom* (descendants of Ishmael [according to Josephus](https://en.wikipedia.org/wiki/Ishmaelites#Josephus '_Antiquities)) were their brothers too, two neighboring peoples in the South that they looked down on? Wouldn't it be more natural that in the discussion of freedom they wanted to emphasize to Jesus of their being the *children of promise* and of their *election* over Esau? Or to take it one step further, since they were from the Jerusalem area, maybe they should have called themselves "**children of Judah**" whose ancestors may have stayed there, not ever been "enslaved" (being exiled to Assyria / Babylon)?