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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 770.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 770.c1 | 770c3 | \([1, -1, 0, -3266669, -2271693567]\) | \(3855131356812007128171561/8967612500\) | \(8967612500\) | \([2]\) | \(10240\) | \(2.0400\) | |
| 770.c2 | 770c4 | \([1, -1, 0, -214949, -31495495]\) | \(1098325674097093229481/205612182617187500\) | \(205612182617187500\) | \([4]\) | \(10240\) | \(2.0400\) | |
| 770.c3 | 770c2 | \([1, -1, 0, -204169, -35456067]\) | \(941226862950447171561/45393906250000\) | \(45393906250000\) | \([2, 2]\) | \(5120\) | \(1.6934\) | |
| 770.c4 | 770c1 | \([1, -1, 0, -12089, -612755]\) | \(-195395722614328041/50730248800000\) | \(-50730248800000\) | \([2]\) | \(2560\) | \(1.3469\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 770.c have rank \(0\).
Complex multiplication
The elliptic curves in class 770.c do not have complex multiplication.Modular form 770.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
