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SageMath
E = EllipticCurve("ga1")
E.isogeny_class()
Elliptic curves in class 388080ga
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.ga4 | 388080ga1 | \([0, 0, 0, -24843, 10939642]\) | \(-4826809/144375\) | \(-50718653314560000\) | \([2]\) | \(2359296\) | \(1.8859\) | \(\Gamma_0(N)\)-optimal |
| 388080.ga3 | 388080ga2 | \([0, 0, 0, -906843, 330752842]\) | \(234770924809/1334025\) | \(468640356626534400\) | \([2, 2]\) | \(4718592\) | \(2.2324\) | |
| 388080.ga1 | 388080ga3 | \([0, 0, 0, -14489643, 21229248922]\) | \(957681397954009/31185\) | \(10955229115944960\) | \([2]\) | \(9437184\) | \(2.5790\) | |
| 388080.ga2 | 388080ga4 | \([0, 0, 0, -1436043, -99698438]\) | \(932288503609/527295615\) | \(185237911629248163840\) | \([2]\) | \(9437184\) | \(2.5790\) |
Rank
sage: E.rank()
The elliptic curves in class 388080ga have rank \(1\).
Complex multiplication
The elliptic curves in class 388080ga do not have complex multiplication.Modular form 388080.2.a.ga
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
