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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 77610ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 77610.bb4 | 77610ba1 | \([1, 0, 0, -408187825, -3104856293143]\) | \(7521489487871336442200725126801/188267790084341760000000000\) | \(188267790084341760000000000\) | \([10]\) | \(35040000\) | \(3.8248\) | \(\Gamma_0(N)\)-optimal |
| 77610.bb3 | 77610ba2 | \([1, 0, 0, -920187825, 6246414106857]\) | \(86169418186840207524668053126801/33010505869278473338982400000\) | \(33010505869278473338982400000\) | \([10]\) | \(70080000\) | \(4.1714\) | |
| 77610.bb2 | 77610ba3 | \([1, 0, 0, -253564675825, -49145236335523543]\) | \(1802976089539026073286824246814593798801/19473767102337600\) | \(19473767102337600\) | \([2]\) | \(175200000\) | \(4.6295\) | |
| 77610.bb1 | 77610ba4 | \([1, 0, 0, -253564676025, -49145236254120303]\) | \(1802976093805336635106536164162943267601/5925431330563846342990380840\) | \(5925431330563846342990380840\) | \([2]\) | \(350400000\) | \(4.9761\) |
Rank
sage: E.rank()
The elliptic curves in class 77610ba have rank \(1\).
Complex multiplication
The elliptic curves in class 77610ba do not have complex multiplication.Modular form 77610.2.a.ba
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
