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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 87360ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 87360.b5 | 87360ec1 | \([0, -1, 0, -23681, -22090719]\) | \(-5602762882081/801531494400\) | \(-210116672067993600\) | \([2]\) | \(1179648\) | \(2.0033\) | \(\Gamma_0(N)\)-optimal |
| 87360.b4 | 87360ec2 | \([0, -1, 0, -1334401, -588059615]\) | \(1002404925316922401/9348917760000\) | \(2450762697277440000\) | \([2, 2]\) | \(2359296\) | \(2.3498\) | |
| 87360.b3 | 87360ec3 | \([0, -1, 0, -2337921, 417266721]\) | \(5391051390768345121/2833965225000000\) | \(742906979942400000000\) | \([2, 2]\) | \(4718592\) | \(2.6964\) | |
| 87360.b2 | 87360ec4 | \([0, -1, 0, -21302401, -37836366815]\) | \(4078208988807294650401/359723582400\) | \(94299378784665600\) | \([2]\) | \(4718592\) | \(2.6964\) | |
| 87360.b6 | 87360ec5 | \([0, -1, 0, 8862079, 3246386721]\) | \(293623352309352854879/187320324116835000\) | \(-49104899045283594240000\) | \([2]\) | \(9437184\) | \(3.0430\) | |
| 87360.b1 | 87360ec6 | \([0, -1, 0, -29594241, 61912975905]\) | \(10934663514379917006241/12996826171875000\) | \(3407040000000000000000\) | \([2]\) | \(9437184\) | \(3.0430\) |
Rank
sage: E.rank()
The elliptic curves in class 87360ec have rank \(0\).
Complex multiplication
The elliptic curves in class 87360ec do not have complex multiplication.Modular form 87360.2.a.ec
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
