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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 3696.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3696.s1 | 3696u4 | \([0, 1, 0, -3624, 82740]\) | \(1285429208617/614922\) | \(2518720512\) | \([2]\) | \(3072\) | \(0.75881\) | |
3696.s2 | 3696u3 | \([0, 1, 0, -2024, -35148]\) | \(223980311017/4278582\) | \(17525071872\) | \([2]\) | \(3072\) | \(0.75881\) | |
3696.s3 | 3696u2 | \([0, 1, 0, -264, 756]\) | \(498677257/213444\) | \(874266624\) | \([2, 2]\) | \(1536\) | \(0.41223\) | |
3696.s4 | 3696u1 | \([0, 1, 0, 56, 116]\) | \(4657463/3696\) | \(-15138816\) | \([2]\) | \(768\) | \(0.065661\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3696.s have rank \(1\).
Complex multiplication
The elliptic curves in class 3696.s do not have complex multiplication.Modular form 3696.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.