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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 97104cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 97104.bf5 | 97104cb1 | \([0, -1, 0, -324808352, -2250216630528]\) | \(38331145780597164097/55468445663232\) | \(5484025587789877915680768\) | \([2]\) | \(26542080\) | \(3.6492\) | \(\Gamma_0(N)\)-optimal |
| 97104.bf4 | 97104cb2 | \([0, -1, 0, -419507872, -831011744000]\) | \(82582985847542515777/44772582831427584\) | \(4426552555117767382234300416\) | \([2, 2]\) | \(53084160\) | \(3.9957\) | |
| 97104.bf6 | 97104cb3 | \([0, -1, 0, 1618011488, -6537695967488]\) | \(4738217997934888496063/2928751705237796928\) | \(-289558308327608257567080579072\) | \([4]\) | \(106168320\) | \(4.3423\) | |
| 97104.bf2 | 97104cb4 | \([0, -1, 0, -3972219552, 95703270024960]\) | \(70108386184777836280897/552468975892674624\) | \(54621216874368083277205733376\) | \([2, 2]\) | \(106168320\) | \(4.3423\) | |
| 97104.bf3 | 97104cb5 | \([0, -1, 0, -1353000992, 220023955131648]\) | \(-2770540998624539614657/209924951154647363208\) | \(-20754751460626146918814356897792\) | \([2]\) | \(212336640\) | \(4.6889\) | |
| 97104.bf1 | 97104cb6 | \([0, -1, 0, -63434824992, 6149519774744832]\) | \(285531136548675601769470657/17941034271597192\) | \(1773784894103723876787191808\) | \([2]\) | \(212336640\) | \(4.6889\) |
Rank
sage: E.rank()
The elliptic curves in class 97104cb have rank \(1\).
Complex multiplication
The elliptic curves in class 97104cb do not have complex multiplication.Modular form 97104.2.a.cb
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 Cremona 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 Cremona labels.
