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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 51870.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51870.g1 | 51870g2 | \([1, 1, 0, -25963, 622807]\) | \(1935594897227176249/946696265563230\) | \(946696265563230\) | \([2]\) | \(353280\) | \(1.5670\) | |
51870.g2 | 51870g1 | \([1, 1, 0, -13813, -623783]\) | \(291498868418706649/3685655528100\) | \(3685655528100\) | \([2]\) | \(176640\) | \(1.2204\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51870.g have rank \(0\).
Complex multiplication
The elliptic curves in class 51870.g do not have complex multiplication.Modular form 51870.2.a.g
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.