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SageMath
E = EllipticCurve("ie1")
E.isogeny_class()
Elliptic curves in class 105840ie
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 105840.if1 | 105840ie1 | \([0, 0, 0, -31752, 2185596]\) | \(-5971968/25\) | \(-14820385708800\) | \([]\) | \(326592\) | \(1.3822\) | \(\Gamma_0(N)\)-optimal |
| 105840.if2 | 105840ie2 | \([0, 0, 0, 74088, 11520684]\) | \(8429568/15625\) | \(-83364669612000000\) | \([]\) | \(979776\) | \(1.9315\) |
Rank
sage: E.rank()
The elliptic curves in class 105840ie have rank \(1\).
Complex multiplication
The elliptic curves in class 105840ie do not have complex multiplication.Modular form 105840.2.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.
