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SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 193200.fr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.fr1 | 193200v4 | \([0, 1, 0, -606596208, 5722308933588]\) | \(385693937170561837203625/2159357734550274048\) | \(138198895011217539072000000\) | \([2]\) | \(99532800\) | \(3.8582\) | |
| 193200.fr2 | 193200v2 | \([0, 1, 0, -44798208, -109973198412]\) | \(155355156733986861625/8291568305839392\) | \(530660371573721088000000\) | \([2]\) | \(33177600\) | \(3.3089\) | |
| 193200.fr3 | 193200v3 | \([0, 1, 0, -16772208, 188580165588]\) | \(-8152944444844179625/235342826399858688\) | \(-15061940889590956032000000\) | \([2]\) | \(49766400\) | \(3.5117\) | |
| 193200.fr4 | 193200v1 | \([0, 1, 0, 1857792, -6863438412]\) | \(11079872671250375/324440155855872\) | \(-20764169974775808000000\) | \([2]\) | \(16588800\) | \(2.9623\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 193200.fr have rank \(1\).
Complex multiplication
The elliptic curves in class 193200.fr do not have complex multiplication.Modular form 193200.2.a.fr
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
