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SageMath
E = EllipticCurve("ia1")
E.isogeny_class()
Elliptic curves in class 169050ia
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169050.e5 | 169050ia1 | \([1, 1, 0, 154325, -47727875]\) | \(221115865823/664731648\) | \(-1221953338368000000\) | \([2]\) | \(3145728\) | \(2.1539\) | \(\Gamma_0(N)\)-optimal |
| 169050.e4 | 169050ia2 | \([1, 1, 0, -1413675, -554191875]\) | \(169967019783457/26337394944\) | \(48415127777604000000\) | \([2, 2]\) | \(6291456\) | \(2.5005\) | |
| 169050.e3 | 169050ia3 | \([1, 1, 0, -6215675, 5424298125]\) | \(14447092394873377/1439452851984\) | \(2646096696610400250000\) | \([2, 2]\) | \(12582912\) | \(2.8471\) | |
| 169050.e2 | 169050ia4 | \([1, 1, 0, -21699675, -38915017875]\) | \(614716917569296417/19093020912\) | \(35098044019935750000\) | \([2]\) | \(12582912\) | \(2.8471\) | |
| 169050.e1 | 169050ia5 | \([1, 1, 0, -96939175, 367320339625]\) | \(54804145548726848737/637608031452\) | \(1172092926442130437500\) | \([2]\) | \(25165824\) | \(3.1936\) | |
| 169050.e6 | 169050ia6 | \([1, 1, 0, 7675825, 26247656625]\) | \(27207619911317663/177609314617308\) | \(-326493097740807326437500\) | \([2]\) | \(25165824\) | \(3.1936\) |
Rank
sage: E.rank()
The elliptic curves in class 169050ia have rank \(1\).
Complex multiplication
The elliptic curves in class 169050ia do not have complex multiplication.Modular form 169050.2.a.ia
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.
