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SageMath
E = EllipticCurve("ia1")
E.isogeny_class()
Elliptic curves in class 92400.ia
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 92400.ia1 | 92400hd6 | \([0, 1, 0, -1219680008, 16394802983988]\) | \(3135316978843283198764801/571725\) | \(36590400000000\) | \([2]\) | \(11796480\) | \(3.3970\) | |
| 92400.ia2 | 92400hd4 | \([0, 1, 0, -76230008, 256149683988]\) | \(765458482133960722801/326869475625\) | \(20919646440000000000\) | \([2, 2]\) | \(5898240\) | \(3.0504\) | |
| 92400.ia3 | 92400hd5 | \([0, 1, 0, -75852008, 258816095988]\) | \(-754127868744065783521/15825714261328125\) | \(-1012845712725000000000000\) | \([2]\) | \(11796480\) | \(3.3970\) | |
| 92400.ia4 | 92400hd3 | \([0, 1, 0, -10178008, -6594196012]\) | \(1821931919215868881/761147600816295\) | \(48713446452242880000000\) | \([2]\) | \(5898240\) | \(3.0504\) | |
| 92400.ia5 | 92400hd2 | \([0, 1, 0, -4788008, 3959423988]\) | \(189674274234120481/3859869269025\) | \(247031633217600000000\) | \([2, 2]\) | \(2949120\) | \(2.7039\) | |
| 92400.ia6 | 92400hd1 | \([0, 1, 0, 13992, 185051988]\) | \(4733169839/231139696095\) | \(-14792940550080000000\) | \([2]\) | \(1474560\) | \(2.3573\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 92400.ia have rank \(1\).
Complex multiplication
The elliptic curves in class 92400.ia do not have complex multiplication.Modular form 92400.2.a.ia
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
