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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 51714h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51714.c2 | 51714h1 | \([1, -1, 0, 144, -324]\) | \(2669927/1836\) | \(-226197036\) | \([]\) | \(23040\) | \(0.29168\) | \(\Gamma_0(N)\)-optimal |
51714.c1 | 51714h2 | \([1, -1, 0, -1611, 30213]\) | \(-3754462153/943296\) | \(-116215010496\) | \([]\) | \(69120\) | \(0.84099\) |
Rank
sage: E.rank()
The elliptic curves in class 51714h have rank \(1\).
Complex multiplication
The elliptic curves in class 51714h do not have complex multiplication.Modular form 51714.2.a.h
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.