Show commands:
SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 303450.ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303450.ce1 | 303450ce8 | \([1, 0, 1, -13877780151, -629258247658052]\) | \(783736670177727068275201/360150\) | \(135830398052343750\) | \([2]\) | \(251658240\) | \(4.0158\) | |
| 303450.ce2 | 303450ce6 | \([1, 0, 1, -867361401, -9832210970552]\) | \(191342053882402567201/129708022500\) | \(48919317858551601562500\) | \([2, 2]\) | \(125829120\) | \(3.6692\) | |
| 303450.ce3 | 303450ce7 | \([1, 0, 1, -861942651, -9961123033052]\) | \(-187778242790732059201/4984939585440150\) | \(-1880067550068640874927343750\) | \([2]\) | \(251658240\) | \(4.0158\) | |
| 303450.ce4 | 303450ce3 | \([1, 0, 1, -108880901, 437166760448]\) | \(378499465220294881/120530818800\) | \(45458139928304643750000\) | \([2]\) | \(62914560\) | \(3.3226\) | |
| 303450.ce5 | 303450ce4 | \([1, 0, 1, -54548901, -151614095552]\) | \(47595748626367201/1215506250000\) | \(458427593426660156250000\) | \([2, 2]\) | \(62914560\) | \(3.3226\) | |
| 303450.ce6 | 303450ce2 | \([1, 0, 1, -7730901, 4851660448]\) | \(135487869158881/51438240000\) | \(19399907300602500000000\) | \([2, 2]\) | \(31457280\) | \(2.9760\) | |
| 303450.ce7 | 303450ce1 | \([1, 0, 1, 1517099, 542092448]\) | \(1023887723039/928972800\) | \(-350361641548800000000\) | \([2]\) | \(15728640\) | \(2.6295\) | \(\Gamma_0(N)\)-optimal |
| 303450.ce8 | 303450ce5 | \([1, 0, 1, 9175599, -484638332552]\) | \(226523624554079/269165039062500\) | \(-101515464105606079101562500\) | \([2]\) | \(125829120\) | \(3.6692\) |
Rank
sage: E.rank()
The elliptic curves in class 303450.ce have rank \(0\).
Complex multiplication
The elliptic curves in class 303450.ce do not have complex multiplication.Modular form 303450.2.a.ce
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
