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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 95550.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 95550.cp1 | 95550bp6 | \([1, 1, 0, -11043400, 14120848750]\) | \(81025909800741361/11088090\) | \(20382854693906250\) | \([2]\) | \(4718592\) | \(2.5417\) | |
| 95550.cp2 | 95550bp4 | \([1, 1, 0, -1035150, -405483000]\) | \(66730743078481/60937500\) | \(112019311523437500\) | \([2]\) | \(2359296\) | \(2.1952\) | |
| 95550.cp3 | 95550bp3 | \([1, 1, 0, -692150, 219120000]\) | \(19948814692561/231344100\) | \(425271906576562500\) | \([2, 2]\) | \(2359296\) | \(2.1952\) | |
| 95550.cp4 | 95550bp5 | \([1, 1, 0, -140900, 559241250]\) | \(-168288035761/73415764890\) | \(-134957676930368906250\) | \([2]\) | \(4718592\) | \(2.5417\) | |
| 95550.cp5 | 95550bp2 | \([1, 1, 0, -79650, -3217500]\) | \(30400540561/15210000\) | \(27960020156250000\) | \([2, 2]\) | \(1179648\) | \(1.8486\) | |
| 95550.cp6 | 95550bp1 | \([1, 1, 0, 18350, -375500]\) | \(371694959/249600\) | \(-458831100000000\) | \([2]\) | \(589824\) | \(1.5020\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 95550.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 95550.cp do not have complex multiplication.Modular form 95550.2.a.cp
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.
