Show commands:
SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 28224.eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28224.eu1 | 28224cb4 | \([0, 0, 0, -37933644, 89925934448]\) | \(268498407453697/252\) | \(5665734653902848\) | \([2]\) | \(1179648\) | \(2.7511\) | |
| 28224.eu2 | 28224cb6 | \([0, 0, 0, -25797324, -49948258192]\) | \(84448510979617/933897762\) | \(20996892513356009177088\) | \([2]\) | \(2359296\) | \(3.0977\) | |
| 28224.eu3 | 28224cb3 | \([0, 0, 0, -2935884, 685259120]\) | \(124475734657/63011844\) | \(1416699953004445433856\) | \([2, 2]\) | \(1179648\) | \(2.7511\) | |
| 28224.eu4 | 28224cb2 | \([0, 0, 0, -2371404, 1404406640]\) | \(65597103937/63504\) | \(1427765132783517696\) | \([2, 2]\) | \(589824\) | \(2.4045\) | |
| 28224.eu5 | 28224cb1 | \([0, 0, 0, -113484, 32494448]\) | \(-7189057/16128\) | \(-362607017849782272\) | \([2]\) | \(294912\) | \(2.0579\) | \(\Gamma_0(N)\)-optimal |
| 28224.eu6 | 28224cb5 | \([0, 0, 0, 10893876, 5293335152]\) | \(6359387729183/4218578658\) | \(-94846616241450678484992\) | \([2]\) | \(2359296\) | \(3.0977\) |
Rank
sage: E.rank()
The elliptic curves in class 28224.eu have rank \(1\).
Complex multiplication
The elliptic curves in class 28224.eu do not have complex multiplication.Modular form 28224.2.a.eu
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
