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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 3234.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3234.l1 | 3234j2 | \([1, 0, 1, -285941, -58875904]\) | \(-448504189023625/135168\) | \(-779216621568\) | \([]\) | \(18144\) | \(1.6460\) | |
3234.l2 | 3234j1 | \([1, 0, 1, -2966, -107656]\) | \(-500313625/574992\) | \(-3314714456592\) | \([3]\) | \(6048\) | \(1.0967\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3234.l have rank \(1\).
Complex multiplication
The elliptic curves in class 3234.l do not have complex multiplication.Modular form 3234.2.a.l
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.