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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 88725s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
88725.cb3 | 88725s1 | \([1, 1, 0, -10650, 396375]\) | \(1771561/105\) | \(7918983515625\) | \([2]\) | \(184320\) | \(1.2281\) | \(\Gamma_0(N)\)-optimal |
88725.cb2 | 88725s2 | \([1, 1, 0, -31775, -1695000]\) | \(47045881/11025\) | \(831493269140625\) | \([2, 2]\) | \(368640\) | \(1.5747\) | |
88725.cb4 | 88725s3 | \([1, 1, 0, 73850, -10461875]\) | \(590589719/972405\) | \(-73337706338203125\) | \([2]\) | \(737280\) | \(1.9212\) | |
88725.cb1 | 88725s4 | \([1, 1, 0, -475400, -126353625]\) | \(157551496201/13125\) | \(989872939453125\) | \([2]\) | \(737280\) | \(1.9212\) |
Rank
sage: E.rank()
The elliptic curves in class 88725s have rank \(0\).
Complex multiplication
The elliptic curves in class 88725s do not have complex multiplication.Modular form 88725.2.a.s
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.