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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 22743b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22743.k2 | 22743b1 | \([1, -1, 0, -3858, 387855]\) | \(-729/7\) | \(-60987974880231\) | \([2]\) | \(48640\) | \(1.3275\) | \(\Gamma_0(N)\)-optimal |
22743.k1 | 22743b2 | \([1, -1, 0, -106743, 13413096]\) | \(15438249/49\) | \(426915824161617\) | \([2]\) | \(97280\) | \(1.6741\) |
Rank
sage: E.rank()
The elliptic curves in class 22743b have rank \(1\).
Complex multiplication
The elliptic curves in class 22743b do not have complex multiplication.Modular form 22743.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.