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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 9702.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9702.e1 | 9702r2 | \([1, -1, 0, -513, -1625]\) | \(59776471/29282\) | \(7321876254\) | \([2]\) | \(6144\) | \(0.58614\) | |
9702.e2 | 9702r1 | \([1, -1, 0, 117, -239]\) | \(704969/484\) | \(-121022748\) | \([2]\) | \(3072\) | \(0.23957\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 9702.e have rank \(1\).
Complex multiplication
The elliptic curves in class 9702.e do not have complex multiplication.Modular form 9702.2.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.