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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 3230.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3230.c1 | 3230e3 | \([1, -1, 1, -126673, -17321253]\) | \(224787763392247177569/1983623750\) | \(1983623750\) | \([2]\) | \(11264\) | \(1.3678\) | |
3230.c2 | 3230e2 | \([1, -1, 1, -7923, -268753]\) | \(54997290038077569/163014062500\) | \(163014062500\) | \([2, 2]\) | \(5632\) | \(1.0212\) | |
3230.c3 | 3230e4 | \([1, -1, 1, -4693, -492269]\) | \(-11428483741113249/98571777343750\) | \(-98571777343750\) | \([2]\) | \(11264\) | \(1.3678\) | |
3230.c4 | 3230e1 | \([1, -1, 1, -703, -169]\) | \(38371643079489/22154570000\) | \(22154570000\) | \([4]\) | \(2816\) | \(0.67463\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3230.c have rank \(0\).
Complex multiplication
The elliptic curves in class 3230.c do not have complex multiplication.Modular form 3230.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.