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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 8624u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8624.z2 | 8624u1 | \([0, -1, 0, -11384, -479632]\) | \(-338608873/13552\) | \(-6530577399808\) | \([2]\) | \(18432\) | \(1.2272\) | \(\Gamma_0(N)\)-optimal |
8624.z1 | 8624u2 | \([0, -1, 0, -183864, -30284176]\) | \(1426487591593/2156\) | \(1038955495424\) | \([2]\) | \(36864\) | \(1.5738\) |
Rank
sage: E.rank()
The elliptic curves in class 8624u have rank \(1\).
Complex multiplication
The elliptic curves in class 8624u do not have complex multiplication.Modular form 8624.2.a.u
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.