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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 182070.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
182070.e1 | 182070de1 | \([1, -1, 0, -95985, -11410659]\) | \(27306250652897/31360000\) | \(112318254720000\) | \([2]\) | \(1105920\) | \(1.6089\) | \(\Gamma_0(N)\)-optimal |
182070.e2 | 182070de2 | \([1, -1, 0, -71505, -17388675]\) | \(-11289171456737/30012500000\) | \(-107492079712500000\) | \([2]\) | \(2211840\) | \(1.9555\) |
Rank
sage: E.rank()
The elliptic curves in class 182070.e have rank \(2\).
Complex multiplication
The elliptic curves in class 182070.e do not have complex multiplication.Modular form 182070.2.a.e
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.