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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 169050.ed
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169050.ed1 | 169050gs4 | \([1, 0, 1, -135240026, -605360479552]\) | \(148809678420065817601/20700\) | \(38052098437500\) | \([2]\) | \(14155776\) | \(2.9307\) | |
| 169050.ed2 | 169050gs5 | \([1, 0, 1, -31641776, 58680268448]\) | \(1905890658841300321/293666194803750\) | \(539836471132287246093750\) | \([2]\) | \(28311552\) | \(3.2773\) | |
| 169050.ed3 | 169050gs3 | \([1, 0, 1, -8673026, -8939731552]\) | \(39248884582600321/3935264062500\) | \(7234060651391601562500\) | \([2, 2]\) | \(14155776\) | \(2.9307\) | |
| 169050.ed4 | 169050gs2 | \([1, 0, 1, -8452526, -9459229552]\) | \(36330796409313601/428490000\) | \(787678437656250000\) | \([2, 2]\) | \(7077888\) | \(2.5841\) | |
| 169050.ed5 | 169050gs1 | \([1, 0, 1, -514526, -155893552]\) | \(-8194759433281/965779200\) | \(-1775358704700000000\) | \([2]\) | \(3538944\) | \(2.2375\) | \(\Gamma_0(N)\)-optimal |
| 169050.ed6 | 169050gs6 | \([1, 0, 1, 10767724, -43310977552]\) | \(75108181893694559/484313964843750\) | \(-890297713279724121093750\) | \([2]\) | \(28311552\) | \(3.2773\) |
Rank
sage: E.rank()
The elliptic curves in class 169050.ed have rank \(0\).
Complex multiplication
The elliptic curves in class 169050.ed do not have complex multiplication.Modular form 169050.2.a.ed
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.
