Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 188790.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 188790.p1 | 188790bj8 | \([1, 0, 1, -98449519809, -11816958471699404]\) | \(105527225205871538162426927151468325129/744380581656383514404296875000000\) | \(744380581656383514404296875000000\) | \([2]\) | \(1182449664\) | \(5.1402\) | |
| 188790.p2 | 188790bj6 | \([1, 0, 1, -10163295489, 85684347653812]\) | \(116098703407847670869343269366467849/64015026145869368241000000000000\) | \(64015026145869368241000000000000\) | \([2, 2]\) | \(591224832\) | \(4.7936\) | |
| 188790.p3 | 188790bj5 | \([1, 0, 1, -7836214494, 256557085748692]\) | \(53216052650416081797407638624167769/2361277490609085356372892187500\) | \(2361277490609085356372892187500\) | \([6]\) | \(394149888\) | \(4.5909\) | |
| 188790.p4 | 188790bj3 | \([1, 0, 1, -7753843969, 262425326890676]\) | \(51555485898049360267782523464060169/84637581633102270431232000000\) | \(84637581633102270431232000000\) | \([2]\) | \(295612416\) | \(4.4470\) | |
| 188790.p5 | 188790bj2 | \([1, 0, 1, -7750790274, 262642638842116]\) | \(51494597620262450539701619974305689/108515066288781742762410000\) | \(108515066288781742762410000\) | \([2, 6]\) | \(197074944\) | \(4.2443\) | |
| 188790.p6 | 188790bj1 | \([1, 0, 1, -7750786354, 262642917792452]\) | \(51494519489369938852805502493805209/228675224461420800\) | \(228675224461420800\) | \([6]\) | \(98537472\) | \(3.8977\) | \(\Gamma_0(N)\)-optimal |
| 188790.p7 | 188790bj4 | \([1, 0, 1, -7665428774, 268710339129716]\) | \(-49811895674016166531252634734409689/2366486214727600599419467464300\) | \(-2366486214727600599419467464300\) | \([6]\) | \(394149888\) | \(4.5909\) | |
| 188790.p8 | 188790bj7 | \([1, 0, 1, 39571704511, 676914133653812]\) | \(6852956649015365077367425396473532151/4163777371027300089658623147000000\) | \(-4163777371027300089658623147000000\) | \([2]\) | \(1182449664\) | \(5.1402\) |
Rank
sage: E.rank()
The elliptic curves in class 188790.p have rank \(1\).
Complex multiplication
The elliptic curves in class 188790.p do not have complex multiplication.Modular form 188790.2.a.p
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
