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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 414736.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414736.p1 | 414736p2 | \([0, -1, 0, -3689422, -2726395741]\) | \(406749952\) | \(13654359055089424\) | \([]\) | \(6286896\) | \(2.3357\) | |
414736.p2 | 414736p1 | \([0, -1, 0, -60482, -1061801]\) | \(1792\) | \(13654359055089424\) | \([]\) | \(2095632\) | \(1.7864\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414736.p have rank \(1\).
Complex multiplication
The elliptic curves in class 414736.p do not have complex multiplication.Modular form 414736.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.