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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 20510.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20510.b1 | 20510a2 | \([1, 0, 1, -623024, 82090366]\) | \(26744657491710624461689/12562375000000000000\) | \(12562375000000000000\) | \([]\) | \(336960\) | \(2.3591\) | |
20510.b2 | 20510a1 | \([1, 0, 1, -519009, 143872932]\) | \(15461341616731869233929/8803814950000\) | \(8803814950000\) | \([3]\) | \(112320\) | \(1.8097\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 20510.b have rank \(1\).
Complex multiplication
The elliptic curves in class 20510.b do not have complex multiplication.Modular form 20510.2.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.