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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 364560ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 364560.ek6 | 364560ek1 | \([0, 1, 0, 47024, -22221676]\) | \(23862997439/457113600\) | \(-220278611666534400\) | \([2]\) | \(4718592\) | \(2.0082\) | \(\Gamma_0(N)\)-optimal |
| 364560.ek5 | 364560ek2 | \([0, 1, 0, -956496, -340538220]\) | \(200828550012481/12454560000\) | \(6001731704586240000\) | \([2, 2]\) | \(9437184\) | \(2.3548\) | |
| 364560.ek4 | 364560ek3 | \([0, 1, 0, -2900816, 1481678484]\) | \(5601911201812801/1271193750000\) | \(612575942630400000000\) | \([2]\) | \(18874368\) | \(2.7014\) | |
| 364560.ek2 | 364560ek4 | \([0, 1, 0, -15068496, -22518957420]\) | \(785209010066844481/3324675600\) | \(1602128935585382400\) | \([2, 2]\) | \(18874368\) | \(2.7014\) | |
| 364560.ek3 | 364560ek5 | \([0, 1, 0, -14833296, -23255697900]\) | \(-749011598724977281/51173462246460\) | \(-24659995278679132323840\) | \([2]\) | \(37748736\) | \(3.0479\) | |
| 364560.ek1 | 364560ek6 | \([0, 1, 0, -241095696, -1440975253740]\) | \(3216206300355197383681/57660\) | \(27785794928640\) | \([2]\) | \(37748736\) | \(3.0479\) |
Rank
sage: E.rank()
The elliptic curves in class 364560ek have rank \(0\).
Complex multiplication
The elliptic curves in class 364560ek do not have complex multiplication.Modular form 364560.2.a.ek
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.
