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Properties

Label 27735.k
Number of curves $8$
Conductor $27735$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 27735.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27735.k1 27735g8 \([1, 0, 1, -3993879, 3071802841]\) \(1114544804970241/405\) \(2560152034845\) \([2]\) \(322560\) \(2.1715\)  
27735.k2 27735g6 \([1, 0, 1, -249654, 47966731]\) \(272223782641/164025\) \(1036861574112225\) \([2, 2]\) \(161280\) \(1.8249\)  
27735.k3 27735g7 \([1, 0, 1, -203429, 66290321]\) \(-147281603041/215233605\) \(-1360569757550061645\) \([2]\) \(322560\) \(2.1715\)  
27735.k4 27735g4 \([1, 0, 1, -147959, -21918073]\) \(56667352321/15\) \(94820445735\) \([2]\) \(80640\) \(1.4783\)  
27735.k5 27735g3 \([1, 0, 1, -18529, 447431]\) \(111284641/50625\) \(320019004355625\) \([2, 2]\) \(80640\) \(1.4783\)  
27735.k6 27735g2 \([1, 0, 1, -9284, -340243]\) \(13997521/225\) \(1422306686025\) \([2, 2]\) \(40320\) \(1.1317\)  
27735.k7 27735g1 \([1, 0, 1, -39, -14819]\) \(-1/15\) \(-94820445735\) \([2]\) \(20160\) \(0.78518\) \(\Gamma_0(N)\)-optimal
27735.k8 27735g5 \([1, 0, 1, 64676, 3376247]\) \(4733169839/3515625\) \(-22223541969140625\) \([2]\) \(161280\) \(1.8249\)  

Rank

sage: E.rank()
 

The elliptic curves in class 27735.k have rank \(1\).

Complex multiplication

The elliptic curves in class 27735.k do not have complex multiplication.

Modular form 27735.2.a.k

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 3 q^{8} + q^{9} - q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - q^{15} - q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.