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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 51714.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51714.l1 | 51714m2 | \([1, -1, 0, -158625, -24273027]\) | \(275602131611533/53934336\) | \(86381933683968\) | \([2]\) | \(442368\) | \(1.6737\) | |
51714.l2 | 51714m1 | \([1, -1, 0, -8865, -461187]\) | \(-48109395853/30081024\) | \(-48178159091712\) | \([2]\) | \(221184\) | \(1.3272\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51714.l have rank \(1\).
Complex multiplication
The elliptic curves in class 51714.l do not have complex multiplication.Modular form 51714.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.