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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 51714.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51714.q1 | 51714u2 | \([1, -1, 1, -211451, 37477041]\) | \(297141543217/7514\) | \(26439840620154\) | \([2]\) | \(387072\) | \(1.6834\) | |
51714.q2 | 51714u1 | \([1, -1, 1, -13721, 541077]\) | \(81182737/11492\) | \(40437403301412\) | \([2]\) | \(193536\) | \(1.3368\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51714.q have rank \(0\).
Complex multiplication
The elliptic curves in class 51714.q do not have complex multiplication.Modular form 51714.2.a.q
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.