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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 11025x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11025.bd3 | 11025x1 | \([1, -1, 0, -27792, 1696491]\) | \(1771561/105\) | \(140710042265625\) | \([2]\) | \(36864\) | \(1.4679\) | \(\Gamma_0(N)\)-optimal |
| 11025.bd2 | 11025x2 | \([1, -1, 0, -82917, -7068384]\) | \(47045881/11025\) | \(14774554437890625\) | \([2, 2]\) | \(73728\) | \(1.8144\) | |
| 11025.bd1 | 11025x3 | \([1, -1, 0, -1240542, -531472509]\) | \(157551496201/13125\) | \(17588755283203125\) | \([2]\) | \(147456\) | \(2.1610\) | |
| 11025.bd4 | 11025x4 | \([1, -1, 0, 192708, -44277759]\) | \(590589719/972405\) | \(-1303115701421953125\) | \([2]\) | \(147456\) | \(2.1610\) |
Rank
sage: E.rank()
The elliptic curves in class 11025x have rank \(1\).
Complex multiplication
The elliptic curves in class 11025x do not have complex multiplication.Modular form 11025.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
