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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 5225.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5225.b1 | 5225c1 | \([0, -1, 1, -683, 8268]\) | \(-2258403328/480491\) | \(-7507671875\) | \([]\) | \(2592\) | \(0.61630\) | \(\Gamma_0(N)\)-optimal |
5225.b2 | 5225c2 | \([0, -1, 1, 4817, -48107]\) | \(790939860992/517504691\) | \(-8086010796875\) | \([]\) | \(7776\) | \(1.1656\) |
Rank
sage: E.rank()
The elliptic curves in class 5225.b have rank \(1\).
Complex multiplication
The elliptic curves in class 5225.b do not have complex multiplication.Modular form 5225.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.