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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 4114b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4114.a4 | 4114b1 | \([1, 0, 1, -366, -1696]\) | \(3048625/1088\) | \(1927458368\) | \([2]\) | \(2160\) | \(0.48256\) | \(\Gamma_0(N)\)-optimal |
| 4114.a3 | 4114b2 | \([1, 0, 1, -5206, -144960]\) | \(8805624625/2312\) | \(4095849032\) | \([2]\) | \(4320\) | \(0.82913\) | |
| 4114.a2 | 4114b3 | \([1, 0, 1, -12466, 534576]\) | \(120920208625/19652\) | \(34814716772\) | \([2]\) | \(6480\) | \(1.0319\) | |
| 4114.a1 | 4114b4 | \([1, 0, 1, -13676, 424224]\) | \(159661140625/48275138\) | \(85522351750418\) | \([2]\) | \(12960\) | \(1.3784\) |
Rank
sage: E.rank()
The elliptic curves in class 4114b have rank \(1\).
Complex multiplication
The elliptic curves in class 4114b do not have complex multiplication.Modular form 4114.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
