Show commands:
SageMath
E = EllipticCurve("lz1")
E.isogeny_class()
Elliptic curves in class 388080.lz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.lz1 | 388080lz5 | \([0, 0, 0, -1207175907, 16143732167906]\) | \(553808571467029327441/12529687500\) | \(4401654555513600000000\) | \([2]\) | \(113246208\) | \(3.6772\) | |
| 388080.lz2 | 388080lz4 | \([0, 0, 0, -83437347, -292709401054]\) | \(182864522286982801/463015182960\) | \(162656322382170104463360\) | \([2]\) | \(56623104\) | \(3.3306\) | |
| 388080.lz3 | 388080lz3 | \([0, 0, 0, -75534627, 251641016354]\) | \(135670761487282321/643043610000\) | \(225899954436246773760000\) | \([2, 2]\) | \(56623104\) | \(3.3306\) | |
| 388080.lz4 | 388080lz6 | \([0, 0, 0, -36726627, 509877209954]\) | \(-15595206456730321/310672490129100\) | \(-109138634259594019721625600\) | \([2]\) | \(113246208\) | \(3.6772\) | |
| 388080.lz5 | 388080lz2 | \([0, 0, 0, -7232547, -707848414]\) | \(119102750067601/68309049600\) | \(23996834666039712153600\) | \([2, 2]\) | \(28311552\) | \(2.9840\) | |
| 388080.lz6 | 388080lz1 | \([0, 0, 0, 1799133, -88275166]\) | \(1833318007919/1070530560\) | \(-376075278512774184960\) | \([2]\) | \(14155776\) | \(2.6374\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 388080.lz have rank \(0\).
Complex multiplication
The elliptic curves in class 388080.lz do not have complex multiplication.Modular form 388080.2.a.lz
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
