Show commands:
SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 44352cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 44352.bj4 | 44352cf1 | \([0, 0, 0, -19596, -1055504]\) | \(4354703137/1617\) | \(309013512192\) | \([2]\) | \(81920\) | \(1.1721\) | \(\Gamma_0(N)\)-optimal |
| 44352.bj3 | 44352cf2 | \([0, 0, 0, -22476, -724880]\) | \(6570725617/2614689\) | \(499674849214464\) | \([2, 2]\) | \(163840\) | \(1.5187\) | |
| 44352.bj6 | 44352cf3 | \([0, 0, 0, 72564, -5248784]\) | \(221115865823/190238433\) | \(-36355130695876608\) | \([2]\) | \(327680\) | \(1.8652\) | |
| 44352.bj2 | 44352cf4 | \([0, 0, 0, -163596, 24958960]\) | \(2533811507137/58110129\) | \(11105018587643904\) | \([2, 2]\) | \(327680\) | \(1.8652\) | |
| 44352.bj5 | 44352cf5 | \([0, 0, 0, 17844, 77286256]\) | \(3288008303/13504609503\) | \(-2580771065741180928\) | \([2]\) | \(655360\) | \(2.2118\) | |
| 44352.bj1 | 44352cf6 | \([0, 0, 0, -2602956, 1616397424]\) | \(10206027697760497/5557167\) | \(1061991151828992\) | \([2]\) | \(655360\) | \(2.2118\) |
Rank
sage: E.rank()
The elliptic curves in class 44352cf have rank \(1\).
Complex multiplication
The elliptic curves in class 44352cf do not have complex multiplication.Modular form 44352.2.a.cf
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.
