Show commands:
SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 457776bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 457776.bj5 | 457776bj1 | \([0, 0, 0, -11819811, 11847493154]\) | \(2533811507137/625016832\) | \(45047709926773827305472\) | \([2]\) | \(28311552\) | \(3.0573\) | \(\Gamma_0(N)\)-optimal |
| 457776.bj4 | 457776bj2 | \([0, 0, 0, -65088291, -192266668510]\) | \(423108074414017/23284318464\) | \(1678203162389226527391744\) | \([2, 2]\) | \(56623104\) | \(3.4038\) | |
| 457776.bj6 | 457776bj3 | \([0, 0, 0, 44777949, -775370750686]\) | \(137763859017023/3683199928848\) | \(-265464405920285206049783808\) | \([2]\) | \(113246208\) | \(3.7504\) | |
| 457776.bj2 | 457776bj4 | \([0, 0, 0, -1027250211, -12672468932830]\) | \(1663303207415737537/5483698704\) | \(395234265536736091766784\) | \([2, 2]\) | \(113246208\) | \(3.7504\) | |
| 457776.bj3 | 457776bj5 | \([0, 0, 0, -1013100771, -13038523435294]\) | \(-1595514095015181697/95635786040388\) | \(-6892891403229288274767003648\) | \([2]\) | \(226492416\) | \(4.0970\) | |
| 457776.bj1 | 457776bj6 | \([0, 0, 0, -16435990371, -811039359346846]\) | \(6812873765474836663297/74052\) | \(5337253086165614592\) | \([2]\) | \(226492416\) | \(4.0970\) |
Rank
sage: E.rank()
The elliptic curves in class 457776bj have rank \(2\).
Complex multiplication
The elliptic curves in class 457776bj do not have complex multiplication.Modular form 457776.2.a.bj
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.
