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SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 243360es
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
243360.es1 | 243360es1 | \([0, 0, 0, -16107897, -24837937436]\) | \(2052450196928704/4317958125\) | \(972402445606592520000\) | \([2]\) | \(12386304\) | \(2.9123\) | \(\Gamma_0(N)\)-optimal |
243360.es2 | 243360es2 | \([0, 0, 0, -10563852, -42193015904]\) | \(-9045718037056/48125390625\) | \(-693620400158337600000000\) | \([2]\) | \(24772608\) | \(3.2588\) |
Rank
sage: E.rank()
The elliptic curves in class 243360es have rank \(1\).
Complex multiplication
The elliptic curves in class 243360es do not have complex multiplication.Modular form 243360.2.a.es
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.