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SageMath
E = EllipticCurve("fp1")
E.isogeny_class()
Elliptic curves in class 77616.fp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 77616.fp1 | 77616gk4 | \([0, 0, 0, -649183899, -6366484463542]\) | \(86129359107301290313/9166294368\) | \(3220101168691226738688\) | \([2]\) | \(17694720\) | \(3.5547\) | |
| 77616.fp2 | 77616gk2 | \([0, 0, 0, -40674459, -98958933430]\) | \(21184262604460873/216872764416\) | \(76186975250461287776256\) | \([2, 2]\) | \(8847360\) | \(3.2081\) | |
| 77616.fp3 | 77616gk3 | \([0, 0, 0, -10192539, -243839499190]\) | \(-333345918055753/72923718045024\) | \(-25617958607337110863478784\) | \([2]\) | \(17694720\) | \(3.5547\) | |
| 77616.fp4 | 77616gk1 | \([0, 0, 0, -4547739, 1234911818]\) | \(29609739866953/15259926528\) | \(5360782151891181109248\) | \([2]\) | \(4423680\) | \(2.8616\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 77616.fp have rank \(1\).
Complex multiplication
The elliptic curves in class 77616.fp do not have complex multiplication.Modular form 77616.2.a.fp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
