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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 174915r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 174915.q4 | 174915r1 | \([1, -1, 0, 694305, -140356800]\) | \(10519294081031/8500170375\) | \(-29909921474468280375\) | \([2]\) | \(4608000\) | \(2.4249\) | \(\Gamma_0(N)\)-optimal |
| 174915.q3 | 174915r2 | \([1, -1, 0, -3328740, -1219337469]\) | \(1159246431432649/488076890625\) | \(1717417513774998140625\) | \([2, 2]\) | \(9216000\) | \(2.7715\) | |
| 174915.q2 | 174915r3 | \([1, -1, 0, -25193115, 47831201406]\) | \(502552788401502649/10024505152875\) | \(35273664963791257462875\) | \([2]\) | \(18432000\) | \(3.1181\) | |
| 174915.q1 | 174915r4 | \([1, -1, 0, -45833085, -119372915700]\) | \(3026030815665395929/1364501953125\) | \(4801332734430908203125\) | \([2]\) | \(18432000\) | \(3.1181\) |
Rank
sage: E.rank()
The elliptic curves in class 174915r have rank \(1\).
Complex multiplication
The elliptic curves in class 174915r do not have complex multiplication.Modular form 174915.2.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 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.
