Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 52983.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 52983.f1 | 52983h4 | \([1, -1, 1, -1548799214, -23460303439834]\) | \(947531277805646290177/38367\) | \(16636936454112303\) | \([2]\) | \(10321920\) | \(3.5236\) | |
| 52983.f2 | 52983h6 | \([1, -1, 1, -321448019, 1803354212756]\) | \(8471112631466271697/1662662681263647\) | \(720974102035793092146419823\) | \([2]\) | \(20643840\) | \(3.8702\) | |
| 52983.f3 | 52983h3 | \([1, -1, 1, -98654504, -351772016542]\) | \(244883173420511137/18418027974129\) | \(7986539500499046805696161\) | \([2, 2]\) | \(10321920\) | \(3.5236\) | |
| 52983.f4 | 52983h2 | \([1, -1, 1, -96800099, -366547915582]\) | \(231331938231569617/1472026689\) | \(638309340934926729201\) | \([2, 2]\) | \(5160960\) | \(3.1770\) | |
| 52983.f5 | 52983h1 | \([1, -1, 1, -5934254, -5955896284]\) | \(-53297461115137/4513839183\) | \(-1957318936889858335647\) | \([4]\) | \(2580480\) | \(2.8305\) | \(\Gamma_0(N)\)-optimal |
| 52983.f6 | 52983h5 | \([1, -1, 1, 94468531, -1561262960140]\) | \(215015459663151503/2552757445339983\) | \(-1106942513120216798702642847\) | \([2]\) | \(20643840\) | \(3.8702\) |
Rank
sage: E.rank()
The elliptic curves in class 52983.f have rank \(1\).
Complex multiplication
The elliptic curves in class 52983.f do not have complex multiplication.Modular form 52983.2.a.f
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.
