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SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 187200.ei
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.ei1 | 187200dg2 | \([0, 0, 0, -180300, 29018000]\) | \(434163602/7605\) | \(11354204160000000\) | \([2]\) | \(1179648\) | \(1.8772\) | |
187200.ei2 | 187200dg1 | \([0, 0, 0, -300, 1298000]\) | \(-4/975\) | \(-727833600000000\) | \([2]\) | \(589824\) | \(1.5306\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.ei have rank \(2\).
Complex multiplication
The elliptic curves in class 187200.ei do not have complex multiplication.Modular form 187200.2.a.ei
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.