Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 44400.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 44400.r1 | 44400x4 | \([0, -1, 0, -136402008, -613122217488]\) | \(4385367890843575421521/24975000000\) | \(1598400000000000000\) | \([2]\) | \(3981312\) | \(3.1047\) | |
| 44400.r2 | 44400x6 | \([0, -1, 0, -121250008, 511696982512]\) | \(3080272010107543650001/15465841417699560\) | \(989813850732771840000000\) | \([2]\) | \(7962624\) | \(3.4513\) | |
| 44400.r3 | 44400x3 | \([0, -1, 0, -11730008, -1732777488]\) | \(2788936974993502801/1593609593601600\) | \(101991013990502400000000\) | \([2, 2]\) | \(3981312\) | \(3.1047\) | |
| 44400.r4 | 44400x2 | \([0, -1, 0, -8530008, -9566377488]\) | \(1072487167529950801/2554882560000\) | \(163512483840000000000\) | \([2, 2]\) | \(1990656\) | \(2.7581\) | |
| 44400.r5 | 44400x1 | \([0, -1, 0, -338008, -260265488]\) | \(-66730743078481/419010969600\) | \(-26816702054400000000\) | \([2]\) | \(995328\) | \(2.4115\) | \(\Gamma_0(N)\)-optimal |
| 44400.r6 | 44400x5 | \([0, -1, 0, 46589992, -13863337488]\) | \(174751791402194852399/102423900876336360\) | \(-6555129656085527040000000\) | \([2]\) | \(7962624\) | \(3.4513\) |
Rank
sage: E.rank()
The elliptic curves in class 44400.r have rank \(0\).
Complex multiplication
The elliptic curves in class 44400.r do not have complex multiplication.Modular form 44400.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
