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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 187200ed
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.hh6 | 187200ed1 | \([0, 0, 0, 215700, 15442000]\) | \(371694959/249600\) | \(-745301606400000000\) | \([2]\) | \(2359296\) | \(2.1181\) | \(\Gamma_0(N)\)-optimal |
| 187200.hh5 | 187200ed2 | \([0, 0, 0, -936300, 128338000]\) | \(30400540561/15210000\) | \(45416816640000000000\) | \([2, 2]\) | \(4718592\) | \(2.4647\) | |
| 187200.hh2 | 187200ed3 | \([0, 0, 0, -12168300, 16324882000]\) | \(66730743078481/60937500\) | \(181958400000000000000\) | \([2]\) | \(9437184\) | \(2.8112\) | |
| 187200.hh3 | 187200ed4 | \([0, 0, 0, -8136300, -8842862000]\) | \(19948814692561/231344100\) | \(690789781094400000000\) | \([2, 2]\) | \(9437184\) | \(2.8112\) | |
| 187200.hh4 | 187200ed5 | \([0, 0, 0, -1656300, -22541582000]\) | \(-168288035761/73415764890\) | \(-219218299309301760000000\) | \([2]\) | \(18874368\) | \(3.1578\) | |
| 187200.hh1 | 187200ed6 | \([0, 0, 0, -129816300, -569300942000]\) | \(81025909800741361/11088090\) | \(33108859330560000000\) | \([2]\) | \(18874368\) | \(3.1578\) |
Rank
sage: E.rank()
The elliptic curves in class 187200ed have rank \(0\).
Complex multiplication
The elliptic curves in class 187200ed do not have complex multiplication.Modular form 187200.2.a.ed
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.
