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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 31117a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31117.d3 | 31117a1 | \([0, -1, 1, -2803, 57616]\) | \(4096000/37\) | \(22008462877\) | \([]\) | \(16128\) | \(0.80712\) | \(\Gamma_0(N)\)-optimal |
31117.d2 | 31117a2 | \([0, -1, 1, -19623, -1018023]\) | \(1404928000/50653\) | \(30129585678613\) | \([]\) | \(48384\) | \(1.3564\) | |
31117.d1 | 31117a3 | \([0, -1, 1, -1575473, -760615110]\) | \(727057727488000/37\) | \(22008462877\) | \([]\) | \(145152\) | \(1.9057\) |
Rank
sage: E.rank()
The elliptic curves in class 31117a have rank \(1\).
Complex multiplication
The elliptic curves in class 31117a do not have complex multiplication.Modular form 31117.2.a.a
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.