Show commands:
SageMath
E = EllipticCurve("bx1")
E.isogeny_class()
Elliptic curves in class 7650.bx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7650.bx1 | 7650ca7 | \([1, -1, 1, -25530755, 49658922497]\) | \(161572377633716256481/914742821250\) | \(10419492448300781250\) | \([2]\) | \(393216\) | \(2.8408\) | |
| 7650.bx2 | 7650ca3 | \([1, -1, 1, -4896005, -4168534003]\) | \(1139466686381936641/4080\) | \(46473750000\) | \([2]\) | \(98304\) | \(2.1476\) | |
| 7650.bx3 | 7650ca5 | \([1, -1, 1, -1624505, 746734997]\) | \(41623544884956481/2962701562500\) | \(33747022485351562500\) | \([2, 2]\) | \(196608\) | \(2.4942\) | |
| 7650.bx4 | 7650ca4 | \([1, -1, 1, -324005, -56974003]\) | \(330240275458561/67652010000\) | \(770598676406250000\) | \([2, 2]\) | \(98304\) | \(2.1476\) | |
| 7650.bx5 | 7650ca2 | \([1, -1, 1, -306005, -65074003]\) | \(278202094583041/16646400\) | \(189612900000000\) | \([2, 2]\) | \(49152\) | \(1.8010\) | |
| 7650.bx6 | 7650ca1 | \([1, -1, 1, -18005, -1138003]\) | \(-56667352321/16711680\) | \(-190356480000000\) | \([2]\) | \(24576\) | \(1.4545\) | \(\Gamma_0(N)\)-optimal |
| 7650.bx7 | 7650ca6 | \([1, -1, 1, 688495, -342499003]\) | \(3168685387909439/6278181696900\) | \(-71512413391251562500\) | \([2]\) | \(196608\) | \(2.4942\) | |
| 7650.bx8 | 7650ca8 | \([1, -1, 1, 1473745, 3256317497]\) | \(31077313442863199/420227050781250\) | \(-4786648750305175781250\) | \([2]\) | \(393216\) | \(2.8408\) |
Rank
sage: E.rank()
The elliptic curves in class 7650.bx have rank \(0\).
Complex multiplication
The elliptic curves in class 7650.bx do not have complex multiplication.Modular form 7650.2.a.bx
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
