Show commands:
SageMath
E = EllipticCurve("fb1")
E.isogeny_class()
Elliptic curves in class 177450fb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
177450.gh7 | 177450fb1 | \([1, 1, 1, 887162, 242726531]\) | \(1023887723039/928972800\) | \(-70062097996800000000\) | \([2]\) | \(7077888\) | \(2.4953\) | \(\Gamma_0(N)\)-optimal |
177450.gh6 | 177450fb2 | \([1, 1, 1, -4520838, 2167974531]\) | \(135487869158881/51438240000\) | \(3879414996502500000000\) | \([2, 2]\) | \(14155776\) | \(2.8419\) | |
177450.gh4 | 177450fb3 | \([1, 1, 1, -63670838, 195470174531]\) | \(378499465220294881/120530818800\) | \(9090300640018893750000\) | \([2]\) | \(28311552\) | \(3.1885\) | |
177450.gh5 | 177450fb4 | \([1, 1, 1, -31898838, -67810193469]\) | \(47595748626367201/1215506250000\) | \(91672132922753906250000\) | \([2, 2]\) | \(28311552\) | \(3.1885\) | |
177450.gh8 | 177450fb5 | \([1, 1, 1, 5365662, -216719135469]\) | \(226523624554079/269165039062500\) | \(-20300128641128540039062500\) | \([2]\) | \(56623104\) | \(3.5350\) | |
177450.gh2 | 177450fb6 | \([1, 1, 1, -507211338, -4396956443469]\) | \(191342053882402567201/129708022500\) | \(9782435162112539062500\) | \([2, 2]\) | \(56623104\) | \(3.5350\) | |
177450.gh3 | 177450fb7 | \([1, 1, 1, -504042588, -4454602343469]\) | \(-187778242790732059201/4984939585440150\) | \(-375958613366543515333593750\) | \([2]\) | \(113246208\) | \(3.8816\) | |
177450.gh1 | 177450fb8 | \([1, 1, 1, -8115380088, -281395164293469]\) | \(783736670177727068275201/360150\) | \(27162113458593750\) | \([2]\) | \(113246208\) | \(3.8816\) |
Rank
sage: E.rank()
The elliptic curves in class 177450fb have rank \(0\).
Complex multiplication
The elliptic curves in class 177450fb do not have complex multiplication.Modular form 177450.2.a.fb
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.