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SageMath
E = EllipticCurve("fo1")
E.isogeny_class()
Elliptic curves in class 129360fo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129360.dh3 | 129360fo1 | \([0, -1, 0, -392800, -82146560]\) | \(13908844989649/1980372240\) | \(954322180766760960\) | \([2]\) | \(1769472\) | \(2.1757\) | \(\Gamma_0(N)\)-optimal |
129360.dh2 | 129360fo2 | \([0, -1, 0, -1662880, 743913472]\) | \(1055257664218129/115307784900\) | \(55565703519027609600\) | \([2, 2]\) | \(3538944\) | \(2.5223\) | |
129360.dh4 | 129360fo3 | \([0, -1, 0, 2217920, 3696426112]\) | \(2503876820718671/13702874328990\) | \(-6603282276070787112960\) | \([2]\) | \(7077888\) | \(2.8689\) | |
129360.dh1 | 129360fo4 | \([0, -1, 0, -25864960, 50638921600]\) | \(3971101377248209009/56495958750\) | \(27224854736808960000\) | \([4]\) | \(7077888\) | \(2.8689\) |
Rank
sage: E.rank()
The elliptic curves in class 129360fo have rank \(1\).
Complex multiplication
The elliptic curves in class 129360fo do not have complex multiplication.Modular form 129360.2.a.fo
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.