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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 26520s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 26520.m6 | 26520s1 | \([0, -1, 0, 1105, -981600]\) | \(9317458724864/26001416731875\) | \(-416022667710000\) | \([4]\) | \(98304\) | \(1.4841\) | \(\Gamma_0(N)\)-optimal |
| 26520.m5 | 26520s2 | \([0, -1, 0, -141700, -20060348]\) | \(1229125878116884816/29018422265625\) | \(7428716100000000\) | \([2, 4]\) | \(196608\) | \(1.8307\) | |
| 26520.m4 | 26520s3 | \([0, -1, 0, -314080, 38479900]\) | \(3346154465291614084/1315155029296875\) | \(1346718750000000000\) | \([4]\) | \(393216\) | \(2.1772\) | |
| 26520.m2 | 26520s4 | \([0, -1, 0, -2254200, -1301925348]\) | \(1237089966354690271204/714574355625\) | \(731724140160000\) | \([2, 2]\) | \(393216\) | \(2.1772\) | |
| 26520.m3 | 26520s5 | \([0, -1, 0, -2241200, -1317696948]\) | \(-607905111321334101602/14874581985380325\) | \(-30463143906058905600\) | \([2]\) | \(786432\) | \(2.5238\) | |
| 26520.m1 | 26520s6 | \([0, -1, 0, -36067200, -83359313748]\) | \(2533559197411478296569602/845325\) | \(1731225600\) | \([2]\) | \(786432\) | \(2.5238\) |
Rank
sage: E.rank()
The elliptic curves in class 26520s have rank \(1\).
Complex multiplication
The elliptic curves in class 26520s do not have complex multiplication.Modular form 26520.2.a.s
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
