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Properties

Label 71610.w
Number of curves $8$
Conductor $71610$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 71610.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
71610.w1 71610v8 \([1, 0, 1, -35793687903, -2606503311941774]\) \(5071577345077405358672070129480413161/1305223540346880\) \(1305223540346880\) \([2]\) \(71663616\) \(4.1710\)  
71610.w2 71610v6 \([1, 0, 1, -2237105503, -40726753738894]\) \(1238178077064700939501913323843561/19863420082571339366400\) \(19863420082571339366400\) \([2, 2]\) \(35831808\) \(3.8244\)  
71610.w3 71610v7 \([1, 0, 1, -2234955103, -40808956649614]\) \(-1234610941444443961535654166441961/4959756025168103384408494080\) \(-4959756025168103384408494080\) \([2]\) \(71663616\) \(4.1710\)  
71610.w4 71610v5 \([1, 0, 1, -441904128, -3575368468994]\) \(9543504578719055764988888125561/607017597894648388602000\) \(607017597894648388602000\) \([6]\) \(23887872\) \(3.6217\)  
71610.w5 71610v3 \([1, 0, 1, -139953503, -635079524494]\) \(303162184157664409258637635561/1210645161377645199360000\) \(1210645161377645199360000\) \([2]\) \(17915904\) \(3.4779\)  
71610.w6 71610v2 \([1, 0, 1, -29294128, -48708276994]\) \(2780131992389059055334685561/584057033586282564000000\) \(584057033586282564000000\) \([2, 6]\) \(11943936\) \(3.2751\)  
71610.w7 71610v1 \([1, 0, 1, -9294128, 10235723006]\) \(88787043140695609254685561/6113889936000000000000\) \(6113889936000000000000\) \([6]\) \(5971968\) \(2.9286\) \(\Gamma_0(N)\)-optimal
71610.w8 71610v4 \([1, 0, 1, 63315872, -294384084994]\) \(28071233668625377622538754439/53684248387892211460602000\) \(-53684248387892211460602000\) \([6]\) \(23887872\) \(3.6217\)  

Rank

sage: E.rank()
 

The elliptic curves in class 71610.w have rank \(0\).

Complex multiplication

The elliptic curves in class 71610.w do not have complex multiplication.

Modular form 71610.2.a.w

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + 2 q^{13} - q^{14} + q^{15} + q^{16} - 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.