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SageMath
E = EllipticCurve("ie1")
E.isogeny_class()
Elliptic curves in class 129360ie
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129360.ia2 | 129360ie1 | \([0, 1, 0, 191280, 15422868]\) | \(668944031/475200\) | \(-549815248178380800\) | \([]\) | \(2032128\) | \(2.0925\) | \(\Gamma_0(N)\)-optimal |
129360.ia1 | 129360ie2 | \([0, 1, 0, -2113680, -1438545900]\) | \(-902612375329/249562500\) | \(-288748459329792000000\) | \([]\) | \(6096384\) | \(2.6418\) |
Rank
sage: E.rank()
The elliptic curves in class 129360ie have rank \(0\).
Complex multiplication
The elliptic curves in class 129360ie do not have complex multiplication.Modular form 129360.2.a.ie
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.