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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 338130.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 338130.bp1 | 338130bp5 | \([1, -1, 0, -23448069, 43708604643]\) | \(81025909800741361/11088090\) | \(195109222803390090\) | \([2]\) | \(18874368\) | \(2.7300\) | |
| 338130.bp2 | 338130bp4 | \([1, -1, 0, -2197899, -1252640295]\) | \(66730743078481/60937500\) | \(1072273787873437500\) | \([2]\) | \(9437184\) | \(2.3834\) | |
| 338130.bp3 | 338130bp3 | \([1, -1, 0, -1469619, 679195233]\) | \(19948814692561/231344100\) | \(4070797364663324100\) | \([2, 2]\) | \(9437184\) | \(2.3834\) | |
| 338130.bp4 | 338130bp6 | \([1, -1, 0, -299169, 1730493423]\) | \(-168288035761/73415764890\) | \(-1291844928134991106890\) | \([2]\) | \(18874368\) | \(2.7300\) | |
| 338130.bp5 | 338130bp2 | \([1, -1, 0, -169119, -9809667]\) | \(30400540561/15210000\) | \(267639537453210000\) | \([2, 2]\) | \(4718592\) | \(2.0368\) | |
| 338130.bp6 | 338130bp1 | \([1, -1, 0, 38961, -1195155]\) | \(371694959/249600\) | \(-4392033435129600\) | \([2]\) | \(2359296\) | \(1.6903\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 338130.bp have rank \(0\).
Complex multiplication
The elliptic curves in class 338130.bp do not have complex multiplication.Modular form 338130.2.a.bp
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.
