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SageMath
E = EllipticCurve("ho1")
E.isogeny_class()
Elliptic curves in class 158400.ho
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 158400.ho1 | 158400ej6 | \([0, 0, 0, -2463624300, -47066274542000]\) | \(553808571467029327441/12529687500\) | \(37413446400000000000000\) | \([2]\) | \(56623104\) | \(3.8555\) | |
| 158400.ho2 | 158400ej3 | \([0, 0, 0, -170280300, 853380178000]\) | \(182864522286982801/463015182960\) | \(1382555928075632640000000\) | \([2]\) | \(28311552\) | \(3.5089\) | |
| 158400.ho3 | 158400ej4 | \([0, 0, 0, -154152300, -733647278000]\) | \(135670761487282321/643043610000\) | \(1920117930762240000000000\) | \([2, 2]\) | \(28311552\) | \(3.5089\) | |
| 158400.ho4 | 158400ej5 | \([0, 0, 0, -74952300, -1486522478000]\) | \(-15595206456730321/310672490129100\) | \(-927663084765650534400000000\) | \([2]\) | \(56623104\) | \(3.8555\) | |
| 158400.ho5 | 158400ej2 | \([0, 0, 0, -14760300, 2063698000]\) | \(119102750067601/68309049600\) | \(203969729160806400000000\) | \([2, 2]\) | \(14155776\) | \(3.1624\) | |
| 158400.ho6 | 158400ej1 | \([0, 0, 0, 3671700, 257362000]\) | \(1833318007919/1070530560\) | \(-3196587123671040000000\) | \([2]\) | \(7077888\) | \(2.8158\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 158400.ho have rank \(1\).
Complex multiplication
The elliptic curves in class 158400.ho do not have complex multiplication.Modular form 158400.2.a.ho
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.
