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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 678.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
678.e1 | 678d2 | \([1, 0, 0, -7121, -2567403]\) | \(-39934705050538129/2823126576537804\) | \(-2823126576537804\) | \([]\) | \(3528\) | \(1.6444\) | |
678.e2 | 678d1 | \([1, 0, 0, -1661, 26097]\) | \(-506814405937489/4048994304\) | \(-4048994304\) | \([7]\) | \(504\) | \(0.67149\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 678.e have rank \(0\).
Complex multiplication
The elliptic curves in class 678.e do not have complex multiplication.Modular form 678.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.