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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 14112z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14112.t3 | 14112z1 | \([0, 0, 0, -6321, -178360]\) | \(5088448/441\) | \(2420662999104\) | \([2, 2]\) | \(24576\) | \(1.1171\) | \(\Gamma_0(N)\)-optimal |
14112.t1 | 14112z2 | \([0, 0, 0, -98931, -11976874]\) | \(2438569736/21\) | \(922157332992\) | \([2]\) | \(49152\) | \(1.4636\) | |
14112.t2 | 14112z3 | \([0, 0, 0, -21756, 1031744]\) | \(3241792/567\) | \(199185983926272\) | \([2]\) | \(49152\) | \(1.4636\) | |
14112.t4 | 14112z4 | \([0, 0, 0, 6909, -826630]\) | \(830584/7203\) | \(-316299965216256\) | \([2]\) | \(49152\) | \(1.4636\) |
Rank
sage: E.rank()
The elliptic curves in class 14112z have rank \(1\).
Complex multiplication
The elliptic curves in class 14112z do not have complex multiplication.Modular form 14112.2.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.