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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 111925d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
111925.o3 | 111925d1 | \([0, -1, 1, -10083, -383307]\) | \(4096000/37\) | \(1024183703125\) | \([]\) | \(129600\) | \(1.1271\) | \(\Gamma_0(N)\)-optimal |
111925.o2 | 111925d2 | \([0, -1, 1, -70583, 7012818]\) | \(1404928000/50653\) | \(1402107489578125\) | \([]\) | \(388800\) | \(1.6764\) | |
111925.o1 | 111925d3 | \([0, -1, 1, -5666833, 5194176943]\) | \(727057727488000/37\) | \(1024183703125\) | \([]\) | \(1166400\) | \(2.2257\) |
Rank
sage: E.rank()
The elliptic curves in class 111925d have rank \(0\).
Complex multiplication
The elliptic curves in class 111925d do not have complex multiplication.Modular form 111925.2.a.d
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.