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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 18810i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18810.h2 | 18810i1 | \([1, -1, 0, -8694, -312012]\) | \(-99697252461409/801155520\) | \(-584042374080\) | \([2]\) | \(36864\) | \(1.0855\) | \(\Gamma_0(N)\)-optimal |
18810.h1 | 18810i2 | \([1, -1, 0, -139374, -19992420]\) | \(410717520667800289/26208600\) | \(19106069400\) | \([2]\) | \(73728\) | \(1.4321\) |
Rank
sage: E.rank()
The elliptic curves in class 18810i have rank \(0\).
Complex multiplication
The elliptic curves in class 18810i do not have complex multiplication.Modular form 18810.2.a.i
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 Cremona 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 Cremona labels.