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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 31824.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31824.e1 | 31824w1 | \([0, 0, 0, -8571, 305386]\) | \(23320116793/2873\) | \(8578732032\) | \([2]\) | \(36864\) | \(0.92943\) | \(\Gamma_0(N)\)-optimal |
31824.e2 | 31824w2 | \([0, 0, 0, -7851, 358810]\) | \(-17923019113/8254129\) | \(-24646697127936\) | \([2]\) | \(73728\) | \(1.2760\) |
Rank
sage: E.rank()
The elliptic curves in class 31824.e have rank \(1\).
Complex multiplication
The elliptic curves in class 31824.e do not have complex multiplication.Modular form 31824.2.a.e
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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.