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SageMath
E = EllipticCurve("hr1")
E.isogeny_class()
Elliptic curves in class 331200.hr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 331200.hr1 | 331200hr3 | \([0, 0, 0, -1589760300, 24397514498000]\) | \(148809678420065817601/20700\) | \(61809868800000000\) | \([2]\) | \(56623104\) | \(3.5468\) | |
| 331200.hr2 | 331200hr6 | \([0, 0, 0, -371952300, -2365109278000]\) | \(1905890658841300321/293666194803750\) | \(876882559024880640000000000\) | \([2]\) | \(113246208\) | \(3.8933\) | |
| 331200.hr3 | 331200hr4 | \([0, 0, 0, -101952300, 360270722000]\) | \(39248884582600321/3935264062500\) | \(11750635526400000000000000\) | \([2, 2]\) | \(56623104\) | \(3.5468\) | |
| 331200.hr4 | 331200hr2 | \([0, 0, 0, -99360300, 381208898000]\) | \(36330796409313601/428490000\) | \(1279464284160000000000\) | \([2, 2]\) | \(28311552\) | \(3.2002\) | |
| 331200.hr5 | 331200hr1 | \([0, 0, 0, -6048300, 6281282000]\) | \(-8194759433281/965779200\) | \(-2883801238732800000000\) | \([2]\) | \(14155776\) | \(2.8536\) | \(\Gamma_0(N)\)-optimal |
| 331200.hr6 | 331200hr5 | \([0, 0, 0, 126575700, 1745607458000]\) | \(75108181893694559/484313964843750\) | \(-1446153750000000000000000000\) | \([2]\) | \(113246208\) | \(3.8933\) |
Rank
sage: E.rank()
The elliptic curves in class 331200.hr have rank \(2\).
Complex multiplication
The elliptic curves in class 331200.hr do not have complex multiplication.Modular form 331200.2.a.hr
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
