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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 86394.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86394.d1 | 86394b6 | \([1, 1, 0, -1659951086, 26030318798796]\) | \(285531136548675601769470657/17941034271597192\) | \(31783636615224993056712\) | \([2]\) | \(39321600\) | \(3.7781\) | |
86394.d2 | 86394b4 | \([1, 1, 0, -103944326, 405065871060]\) | \(70108386184777836280897/552468975892674624\) | \(978732491401402549568064\) | \([2, 2]\) | \(19660800\) | \(3.4315\) | |
86394.d3 | 86394b5 | \([1, 1, 0, -35405086, 931351279324]\) | \(-2770540998624539614657/209924951154647363208\) | \(-371894856392478237412127688\) | \([2]\) | \(39321600\) | \(3.7781\) | |
86394.d4 | 86394b2 | \([1, 1, 0, -10977606, -3522863340]\) | \(82582985847542515777/44772582831427584\) | \(79317361613426682138624\) | \([2, 2]\) | \(9830400\) | \(3.0849\) | |
86394.d5 | 86394b1 | \([1, 1, 0, -8499526, -9529233644]\) | \(38331145780597164097/55468445663232\) | \(98265735067600945152\) | \([2]\) | \(4915200\) | \(2.7384\) | \(\Gamma_0(N)\)-optimal |
86394.d6 | 86394b3 | \([1, 1, 0, 42339834, -27654336684]\) | \(4738217997934888496063/2928751705237796928\) | \(-5188462299682776763564608\) | \([2]\) | \(19660800\) | \(3.4315\) |
Rank
sage: E.rank()
The elliptic curves in class 86394.d have rank \(0\).
Complex multiplication
The elliptic curves in class 86394.d do not have complex multiplication.Modular form 86394.2.a.d
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.