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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 22050b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.l2 | 22050b1 | \([1, -1, 0, 1608, -1697984]\) | \(189/512\) | \(-1245197016000000\) | \([]\) | \(108864\) | \(1.5755\) | \(\Gamma_0(N)\)-optimal |
22050.l1 | 22050b2 | \([1, -1, 0, -1027392, -400606984]\) | \(-67645179/8\) | \(-14183572260375000\) | \([]\) | \(326592\) | \(2.1248\) |
Rank
sage: E.rank()
The elliptic curves in class 22050b have rank \(1\).
Complex multiplication
The elliptic curves in class 22050b do not have complex multiplication.Modular form 22050.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.