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SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 163170ep
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
163170.y2 | 163170ep1 | \([1, -1, 0, -155816775, 750785089725]\) | \(-14221861969864791943/46510217625600\) | \(-1368227326746445008076800\) | \([2]\) | \(45416448\) | \(3.4969\) | \(\Gamma_0(N)\)-optimal |
163170.y1 | 163170ep2 | \([1, -1, 0, -2495021895, 47969511801021]\) | \(58389789169255064704903/621457920000\) | \(18281912060953013760000\) | \([2]\) | \(90832896\) | \(3.8435\) |
Rank
sage: E.rank()
The elliptic curves in class 163170ep have rank \(0\).
Complex multiplication
The elliptic curves in class 163170ep do not have complex multiplication.Modular form 163170.2.a.ep
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.