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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 20160ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20160.cz7 | 20160ce1 | \([0, 0, 0, 120948, -12199696]\) | \(1023887723039/928972800\) | \(-177529466703052800\) | \([2]\) | \(196608\) | \(1.9972\) | \(\Gamma_0(N)\)-optimal |
| 20160.cz6 | 20160ce2 | \([0, 0, 0, -616332, -109225744]\) | \(135487869158881/51438240000\) | \(9830000744202240000\) | \([2, 2]\) | \(393216\) | \(2.3437\) | |
| 20160.cz4 | 20160ce3 | \([0, 0, 0, -8680332, -9840860944]\) | \(378499465220294881/120530818800\) | \(23033798172396748800\) | \([2]\) | \(786432\) | \(2.6903\) | |
| 20160.cz5 | 20160ce4 | \([0, 0, 0, -4348812, 3412742384]\) | \(47595748626367201/1215506250000\) | \(232286861721600000000\) | \([2, 2]\) | \(786432\) | \(2.6903\) | |
| 20160.cz2 | 20160ce5 | \([0, 0, 0, -69148812, 221322182384]\) | \(191342053882402567201/129708022500\) | \(24787589110824960000\) | \([2, 2]\) | \(1572864\) | \(3.0369\) | |
| 20160.cz8 | 20160ce6 | \([0, 0, 0, 731508, 10909262576]\) | \(226523624554079/269165039062500\) | \(-51438240000000000000000\) | \([2]\) | \(1572864\) | \(3.0369\) | |
| 20160.cz1 | 20160ce7 | \([0, 0, 0, -1106380812, 14164624511984]\) | \(783736670177727068275201/360150\) | \(68825736806400\) | \([4]\) | \(3145728\) | \(3.3835\) | |
| 20160.cz3 | 20160ce8 | \([0, 0, 0, -68716812, 224224012784]\) | \(-187778242790732059201/4984939585440150\) | \(-952636789957818934886400\) | \([2]\) | \(3145728\) | \(3.3835\) |
Rank
sage: E.rank()
The elliptic curves in class 20160ce have rank \(1\).
Complex multiplication
The elliptic curves in class 20160ce do not have complex multiplication.Modular form 20160.2.a.ce
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
