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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 53312.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53312.s1 | 53312cj2 | \([0, 1, 0, -3201, -48833]\) | \(941192/289\) | \(1114130382848\) | \([2]\) | \(110592\) | \(1.0163\) | |
53312.s2 | 53312cj1 | \([0, 1, 0, -1241, 15847]\) | \(438976/17\) | \(8192135168\) | \([2]\) | \(55296\) | \(0.66974\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 53312.s have rank \(0\).
Complex multiplication
The elliptic curves in class 53312.s do not have complex multiplication.Modular form 53312.2.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 LMFDB 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 LMFDB labels.