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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 85698m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85698.s3 | 85698m1 | \([1, -1, 1, -2480, -49189]\) | \(-140625/8\) | \(-95927256072\) | \([]\) | \(71280\) | \(0.86416\) | \(\Gamma_0(N)\)-optimal |
| 85698.s4 | 85698m2 | \([1, -1, 1, 13390, -95741]\) | \(3375/2\) | \(-157344681772098\) | \([]\) | \(213840\) | \(1.4135\) | |
| 85698.s2 | 85698m3 | \([1, -1, 1, -50090, 8777705]\) | \(-1159088625/2097152\) | \(-25146754615738368\) | \([]\) | \(498960\) | \(1.8371\) | |
| 85698.s1 | 85698m4 | \([1, -1, 1, -5128490, 4471540201]\) | \(-189613868625/128\) | \(-10070059633414272\) | \([]\) | \(1496880\) | \(2.3864\) |
Rank
sage: E.rank()
The elliptic curves in class 85698m have rank \(1\).
Complex multiplication
The elliptic curves in class 85698m do not have complex multiplication.Modular form 85698.2.a.m
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
