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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 212940.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212940.w1 | 212940z1 | \([0, 0, 0, -6177288, -3929068663]\) | \(463030539649024/149501953125\) | \(8416945037054531250000\) | \([2]\) | \(14515200\) | \(2.9105\) | \(\Gamma_0(N)\)-optimal |
212940.w2 | 212940z2 | \([0, 0, 0, 17588337, -26853390538]\) | \(667990736021936/732392128125\) | \(-659737659250011232800000\) | \([2]\) | \(29030400\) | \(3.2571\) |
Rank
sage: E.rank()
The elliptic curves in class 212940.w have rank \(0\).
Complex multiplication
The elliptic curves in class 212940.w do not have complex multiplication.Modular form 212940.2.a.w
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.