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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 123200fk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123200.cj1 | 123200fk1 | \([0, -1, 0, -8933, -322013]\) | \(-78843215872/539\) | \(-539000000\) | \([]\) | \(103680\) | \(0.85629\) | \(\Gamma_0(N)\)-optimal |
| 123200.cj2 | 123200fk2 | \([0, -1, 0, -4933, -615013]\) | \(-13278380032/156590819\) | \(-156590819000000\) | \([]\) | \(311040\) | \(1.4056\) | |
| 123200.cj3 | 123200fk3 | \([0, -1, 0, 44067, 15946987]\) | \(9463555063808/115539436859\) | \(-115539436859000000\) | \([]\) | \(933120\) | \(1.9549\) |
Rank
sage: E.rank()
The elliptic curves in class 123200fk have rank \(1\).
Complex multiplication
The elliptic curves in class 123200fk do not have complex multiplication.Modular form 123200.2.a.fk
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
