Properties

Label 67760cg
Number of curves $4$
Conductor $67760$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cg1")
 
E.isogeny_class()
 

Elliptic curves in class 67760cg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
67760.bo4 67760cg1 \([0, 0, 0, -56507, -54712086]\) \(-2749884201/176619520\) \(-1281606670216069120\) \([2]\) \(737280\) \(2.1543\) \(\Gamma_0(N)\)-optimal
67760.bo3 67760cg2 \([0, 0, 0, -2534587, -1543046934]\) \(248158561089321/1859334400\) \(13491913969657446400\) \([2, 2]\) \(1474560\) \(2.5009\)  
67760.bo2 67760cg3 \([0, 0, 0, -4238267, 792016874]\) \(1160306142246441/634128110000\) \(4601432591072092160000\) \([4]\) \(2949120\) \(2.8475\)  
67760.bo1 67760cg4 \([0, 0, 0, -40480187, -99131541014]\) \(1010962818911303721/57392720\) \(416459589369528320\) \([2]\) \(2949120\) \(2.8475\)  

Rank

sage: E.rank()
 

The elliptic curves in class 67760cg have rank \(1\).

Complex multiplication

The elliptic curves in class 67760cg do not have complex multiplication.

Modular form 67760.2.a.cg

sage: E.q_eigenform(10)
 
\(q + q^{5} + q^{7} - 3 q^{9} + 6 q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 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.