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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 84525.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84525.r1 | 84525ca4 | \([1, 0, 0, -5259563, -4643159508]\) | \(8753151307882969/65205\) | \(119864110078125\) | \([2]\) | \(1622016\) | \(2.2960\) | |
| 84525.r2 | 84525ca2 | \([1, 0, 0, -328938, -72470133]\) | \(2141202151369/5832225\) | \(10721178734765625\) | \([2, 2]\) | \(811008\) | \(1.9494\) | |
| 84525.r3 | 84525ca3 | \([1, 0, 0, -200313, -129708258]\) | \(-483551781049/3672913125\) | \(-6751789941298828125\) | \([2]\) | \(1622016\) | \(2.2960\) | |
| 84525.r4 | 84525ca1 | \([1, 0, 0, -28813, -140008]\) | \(1439069689/828345\) | \(1522718139140625\) | \([2]\) | \(405504\) | \(1.6029\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 84525.r have rank \(1\).
Complex multiplication
The elliptic curves in class 84525.r do not have complex multiplication.Modular form 84525.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
