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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 137677b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
137677.b3 | 137677b1 | \([0, 1, 1, -12403, 523386]\) | \(4096000/37\) | \(1906253851357\) | \([]\) | \(151200\) | \(1.1789\) | \(\Gamma_0(N)\)-optimal |
137677.b2 | 137677b2 | \([0, 1, 1, -86823, -9564245]\) | \(1404928000/50653\) | \(2609661522507733\) | \([]\) | \(453600\) | \(1.7282\) | |
137677.b1 | 137677b3 | \([0, 1, 1, -6970673, -7086024368]\) | \(727057727488000/37\) | \(1906253851357\) | \([]\) | \(1360800\) | \(2.2775\) |
Rank
sage: E.rank()
The elliptic curves in class 137677b have rank \(1\).
Complex multiplication
The elliptic curves in class 137677b do not have complex multiplication.Modular form 137677.2.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 Cremona numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.