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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 31824.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31824.q1 | 31824c2 | \([0, 0, 0, -756435, 129974114]\) | \(64122592551794500/27331783704693\) | \(20403067208418505728\) | \([2]\) | \(589824\) | \(2.4018\) | |
31824.q2 | 31824c1 | \([0, 0, 0, -360975, -82071538]\) | \(27873248949250000/538367795433\) | \(100472351454888192\) | \([2]\) | \(294912\) | \(2.0552\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 31824.q have rank \(0\).
Complex multiplication
The elliptic curves in class 31824.q do not have complex multiplication.Modular form 31824.2.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.