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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 269790bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 269790.bb6 | 269790bb1 | \([1, 1, 1, -42331, -4136407]\) | \(-56667352321/16711680\) | \(-2473928405483520\) | \([2]\) | \(1622016\) | \(1.6682\) | \(\Gamma_0(N)\)-optimal |
| 269790.bb5 | 269790bb2 | \([1, 1, 1, -719451, -235169751]\) | \(278202094583041/16646400\) | \(2464264622649600\) | \([2, 2]\) | \(3244032\) | \(2.0148\) | |
| 269790.bb4 | 269790bb3 | \([1, 1, 1, -761771, -206002807]\) | \(330240275458561/67652010000\) | \(10014925442986890000\) | \([2, 2]\) | \(6488064\) | \(2.3613\) | |
| 269790.bb2 | 269790bb4 | \([1, 1, 1, -11511051, -15036928311]\) | \(1139466686381936641/4080\) | \(603986427120\) | \([2]\) | \(6488064\) | \(2.3613\) | |
| 269790.bb3 | 269790bb5 | \([1, 1, 1, -3819391, 2688951809]\) | \(41623544884956481/2962701562500\) | \(438586159646376562500\) | \([2, 2]\) | \(12976128\) | \(2.7079\) | |
| 269790.bb7 | 269790bb6 | \([1, 1, 1, 1618729, -1233426607]\) | \(3168685387909439/6278181696900\) | \(-929396208804120044100\) | \([2]\) | \(12976128\) | \(2.7079\) | |
| 269790.bb1 | 269790bb7 | \([1, 1, 1, -60025641, 178974234309]\) | \(161572377633716256481/914742821250\) | \(135414766750111841250\) | \([2]\) | \(25952256\) | \(3.0545\) | |
| 269790.bb8 | 269790bb8 | \([1, 1, 1, 3464939, 11741917133]\) | \(31077313442863199/420227050781250\) | \(-62208685044250488281250\) | \([2]\) | \(25952256\) | \(3.0545\) |
Rank
sage: E.rank()
The elliptic curves in class 269790bb have rank \(1\).
Complex multiplication
The elliptic curves in class 269790bb do not have complex multiplication.Modular form 269790.2.a.bb
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
