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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 117600z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117600.dg3 | 117600z1 | \([0, -1, 0, -17558, -819888]\) | \(5088448/441\) | \(51883209000000\) | \([2, 2]\) | \(393216\) | \(1.3725\) | \(\Gamma_0(N)\)-optimal |
117600.dg4 | 117600z2 | \([0, -1, 0, 19192, -3833388]\) | \(830584/7203\) | \(-6779405976000000\) | \([2]\) | \(786432\) | \(1.7190\) | |
117600.dg2 | 117600z3 | \([0, -1, 0, -60433, 4796737]\) | \(3241792/567\) | \(4269246912000000\) | \([2]\) | \(786432\) | \(1.7190\) | |
117600.dg1 | 117600z4 | \([0, -1, 0, -274808, -55356888]\) | \(2438569736/21\) | \(19765032000000\) | \([2]\) | \(786432\) | \(1.7190\) |
Rank
sage: E.rank()
The elliptic curves in class 117600z have rank \(0\).
Complex multiplication
The elliptic curves in class 117600z do not have complex multiplication.Modular form 117600.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.