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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 82800cs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
82800.k2 | 82800cs1 | \([0, 0, 0, -3675, 74250]\) | \(3176523/460\) | \(794880000000\) | \([2]\) | \(110592\) | \(1.0087\) | \(\Gamma_0(N)\)-optimal |
82800.k1 | 82800cs2 | \([0, 0, 0, -15675, -681750]\) | \(246491883/26450\) | \(45705600000000\) | \([2]\) | \(221184\) | \(1.3553\) |
Rank
sage: E.rank()
The elliptic curves in class 82800cs have rank \(1\).
Complex multiplication
The elliptic curves in class 82800cs do not have complex multiplication.Modular form 82800.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.