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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 51800.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51800.m1 | 51800b1 | \([0, 0, 0, -422050, -105534375]\) | \(33256413948450816/2481997\) | \(620499250000\) | \([2]\) | \(195840\) | \(1.7125\) | \(\Gamma_0(N)\)-optimal |
51800.m2 | 51800b2 | \([0, 0, 0, -421175, -105993750]\) | \(-2065624967846736/17960084863\) | \(-71840339452000000\) | \([2]\) | \(391680\) | \(2.0590\) |
Rank
sage: E.rank()
The elliptic curves in class 51800.m have rank \(1\).
Complex multiplication
The elliptic curves in class 51800.m do not have complex multiplication.Modular form 51800.2.a.m
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.