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SageMath
E = EllipticCurve("cx1")
E.isogeny_class()
Elliptic curves in class 111090cx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111090.da7 | 111090cx1 | \([1, 0, 0, 111079, -10728135]\) | \(1023887723039/928972800\) | \(-137521314304819200\) | \([2]\) | \(1441792\) | \(1.9759\) | \(\Gamma_0(N)\)-optimal |
| 111090.da6 | 111090cx2 | \([1, 0, 0, -566041, -96180679]\) | \(135487869158881/51438240000\) | \(7614705586995360000\) | \([2, 2]\) | \(2883584\) | \(2.3225\) | |
| 111090.da5 | 111090cx3 | \([1, 0, 0, -3993961, 3003344585]\) | \(47595748626367201/1215506250000\) | \(179938548303806250000\) | \([2, 2]\) | \(5767168\) | \(2.6690\) | |
| 111090.da4 | 111090cx4 | \([1, 0, 0, -7972041, -8661960279]\) | \(378499465220294881/120530818800\) | \(17842886912955913200\) | \([2]\) | \(5767168\) | \(2.6690\) | |
| 111090.da8 | 111090cx5 | \([1, 0, 0, 671819, 9601690661]\) | \(226523624554079/269165039062500\) | \(-39846085845336914062500\) | \([2]\) | \(11534336\) | \(3.0156\) | |
| 111090.da2 | 111090cx6 | \([1, 0, 0, -63506461, 194788327085]\) | \(191342053882402567201/129708022500\) | \(19201442421219502500\) | \([2, 2]\) | \(11534336\) | \(3.0156\) | |
| 111090.da3 | 111090cx7 | \([1, 0, 0, -63109711, 197342365535]\) | \(-187778242790732059201/4984939585440150\) | \(-737949963141924061543350\) | \([2]\) | \(23068672\) | \(3.3622\) | |
| 111090.da1 | 111090cx8 | \([1, 0, 0, -1016103211, 12466711218635]\) | \(783736670177727068275201/360150\) | \(53315125423350\) | \([2]\) | \(23068672\) | \(3.3622\) |
Rank
sage: E.rank()
The elliptic curves in class 111090cx have rank \(1\).
Complex multiplication
The elliptic curves in class 111090cx do not have complex multiplication.Modular form 111090.2.a.cx
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
