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SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 124950de
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 124950.cp5 | 124950de1 | \([1, 0, 1, -41676, -3040502]\) | \(4354703137/352512\) | \(648010692000000\) | \([2]\) | \(786432\) | \(1.5850\) | \(\Gamma_0(N)\)-optimal |
| 124950.cp4 | 124950de2 | \([1, 0, 1, -139676, 16559498]\) | \(163936758817/30338064\) | \(55769420180250000\) | \([2, 2]\) | \(1572864\) | \(1.9316\) | |
| 124950.cp6 | 124950de3 | \([1, 0, 1, 276824, 96527498]\) | \(1276229915423/2927177028\) | \(-5380928908862062500\) | \([2]\) | \(3145728\) | \(2.2782\) | |
| 124950.cp2 | 124950de4 | \([1, 0, 1, -2124176, 1191383498]\) | \(576615941610337/27060804\) | \(49744945778062500\) | \([2, 2]\) | \(3145728\) | \(2.2782\) | |
| 124950.cp3 | 124950de5 | \([1, 0, 1, -2013926, 1320596498]\) | \(-491411892194497/125563633938\) | \(-230819312018308781250\) | \([2]\) | \(6291456\) | \(2.6248\) | |
| 124950.cp1 | 124950de6 | \([1, 0, 1, -33986426, 76258844498]\) | \(2361739090258884097/5202\) | \(9562657781250\) | \([2]\) | \(6291456\) | \(2.6248\) |
Rank
sage: E.rank()
The elliptic curves in class 124950de have rank \(2\).
Complex multiplication
The elliptic curves in class 124950de do not have complex multiplication.Modular form 124950.2.a.de
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
