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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 5040.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.b1 | 5040a1 | \([0, 0, 0, -318, -2133]\) | \(8232302592/214375\) | \(92610000\) | \([2]\) | \(1536\) | \(0.31038\) | \(\Gamma_0(N)\)-optimal |
5040.b2 | 5040a2 | \([0, 0, 0, 57, -6858]\) | \(2963088/2941225\) | \(-20329747200\) | \([2]\) | \(3072\) | \(0.65695\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.b have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.b do not have complex multiplication.Modular form 5040.2.a.b
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.