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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 121680.cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121680.cs1 | 121680dy4 | \([0, 0, 0, -2105758083, 37192967289218]\) | \(71647584155243142409/10140000\) | \(146145532872867840000\) | \([2]\) | \(41287680\) | \(3.7222\) | |
| 121680.cs2 | 121680dy3 | \([0, 0, 0, -151090563, 397840199042]\) | \(26465989780414729/10571870144160\) | \(152369979850199738719272960\) | \([2]\) | \(41287680\) | \(3.7222\) | |
| 121680.cs3 | 121680dy2 | \([0, 0, 0, -131621763, 581029925762]\) | \(17496824387403529/6580454400\) | \(94842605013176313446400\) | \([2, 2]\) | \(20643840\) | \(3.3756\) | |
| 121680.cs4 | 121680dy1 | \([0, 0, 0, -7021443, 11830743938]\) | \(-2656166199049/2658140160\) | \(-38311174569425067048960\) | \([2]\) | \(10321920\) | \(3.0290\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121680.cs have rank \(1\).
Complex multiplication
The elliptic curves in class 121680.cs do not have complex multiplication.Modular form 121680.2.a.cs
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.
