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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 290400.ec
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
290400.ec1 | 290400ec4 | \([0, 1, 0, -10894033, -13843467937]\) | \(1261112198464/675\) | \(76531435200000000\) | \([2]\) | \(11796480\) | \(2.5693\) | |
290400.ec2 | 290400ec2 | \([0, 1, 0, -1501408, 393709688]\) | \(26410345352/10546875\) | \(149475459375000000000\) | \([2]\) | \(11796480\) | \(2.5693\) | |
290400.ec3 | 290400ec1 | \([0, 1, 0, -684658, -213952312]\) | \(20034997696/455625\) | \(807167480625000000\) | \([2, 2]\) | \(5898240\) | \(2.2227\) | \(\Gamma_0(N)\)-optimal |
290400.ec4 | 290400ec3 | \([0, 1, 0, 71592, -660139812]\) | \(2863288/13286025\) | \(-188296029880200000000\) | \([2]\) | \(11796480\) | \(2.5693\) |
Rank
sage: E.rank()
The elliptic curves in class 290400.ec have rank \(0\).
Complex multiplication
The elliptic curves in class 290400.ec do not have complex multiplication.Modular form 290400.2.a.ec
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.