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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 440818z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
440818.z2 | 440818z1 | \([1, 1, 1, -24853539, 47681283137]\) | \(-661725825335468713/20764291072\) | \(-53275489967593320448\) | \([2]\) | \(30468096\) | \(2.8813\) | \(\Gamma_0(N)\)-optimal* |
440818.z1 | 440818z2 | \([1, 1, 1, -397659619, 3052050920641]\) | \(2710490216437522251433/2668736\) | \(6847246433849024\) | \([2]\) | \(60936192\) | \(3.2279\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 440818z have rank \(1\).
Complex multiplication
The elliptic curves in class 440818z do not have complex multiplication.Modular form 440818.2.a.z
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 Cremona 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 Cremona labels.