Properties

Label 20160.ek
Number of curves $4$
Conductor $20160$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("ek1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 20160.ek

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
20160.ek1 20160fi3 \([0, 0, 0, -3674160012, -85720602100016]\) \(229625675762164624948320008/9568125\) \(228562145280000\) \([2]\) \(6881280\) \(3.6559\)  
20160.ek2 20160fi2 \([0, 0, 0, -229635012, -1339384270016]\) \(448487713888272974160064/91549016015625\) \(273363897038400000000\) \([2, 2]\) \(3440640\) \(3.3093\)  
20160.ek3 20160fi4 \([0, 0, 0, -228847692, -1349024531024]\) \(-55486311952875723077768/801237030029296875\) \(-19139847615000000000000000\) \([4]\) \(6881280\) \(3.6559\)  
20160.ek4 20160fi1 \([0, 0, 0, -14401407, -20777112344]\) \(7079962908642659949376/100085966990454375\) \(4669610875906639320000\) \([2]\) \(1720320\) \(2.9627\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 20160.ek have rank \(1\).

Complex multiplication

The elliptic curves in class 20160.ek do not have complex multiplication.

Modular form 20160.2.a.ek

sage: E.q_eigenform(10)
 
\(q + q^{5} + q^{7} - 4q^{11} + 6q^{13} - 6q^{17} + 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.