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SageMath
E = EllipticCurve("gs1")
E.isogeny_class()
Elliptic curves in class 305760.gs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 305760.gs1 | 305760gs4 | \([0, 1, 0, -51025, -4453345]\) | \(30488290624/195\) | \(93968609280\) | \([2]\) | \(589824\) | \(1.2902\) | |
| 305760.gs2 | 305760gs2 | \([0, 1, 0, -10600, 338120]\) | \(2186875592/428415\) | \(25806129323520\) | \([2]\) | \(589824\) | \(1.2902\) | |
| 305760.gs3 | 305760gs1 | \([0, 1, 0, -3250, -67600]\) | \(504358336/38025\) | \(286310606400\) | \([2, 2]\) | \(294912\) | \(0.94358\) | \(\Gamma_0(N)\)-optimal |
| 305760.gs4 | 305760gs3 | \([0, 1, 0, 3120, -294372]\) | \(55742968/658125\) | \(-39643007040000\) | \([2]\) | \(589824\) | \(1.2902\) |
Rank
sage: E.rank()
The elliptic curves in class 305760.gs have rank \(1\).
Complex multiplication
The elliptic curves in class 305760.gs do not have complex multiplication.Modular form 305760.2.a.gs
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.
