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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 289289.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 289289.d1 | 289289d4 | \([1, -1, 1, -2865634, -1866433162]\) | \(107818231938348177/4463459\) | \(107737049591171\) | \([2]\) | \(3112960\) | \(2.1788\) | |
| 289289.d2 | 289289d3 | \([1, -1, 1, -290644, 11376706]\) | \(112489728522417/62811265517\) | \(1516111255393908173\) | \([2]\) | \(3112960\) | \(2.1788\) | |
| 289289.d3 | 289289d2 | \([1, -1, 1, -179379, -29034742]\) | \(26444947540257/169338169\) | \(4087411738571161\) | \([2, 2]\) | \(1556480\) | \(1.8323\) | |
| 289289.d4 | 289289d1 | \([1, -1, 1, -4534, -989604]\) | \(-426957777/17320303\) | \(-418070008763407\) | \([2]\) | \(778240\) | \(1.4857\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 289289.d have rank \(0\).
Complex multiplication
The elliptic curves in class 289289.d do not have complex multiplication.Modular form 289289.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
