Properties

Label 106722gv
Number of curves $4$
Conductor $106722$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("gv1")
 
E.isogeny_class()
 

Elliptic curves in class 106722gv

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
106722.gs4 106722gv1 \([1, -1, 1, -5177129, -16578743079]\) \(-100999381393/723148272\) \(-109875087041458428475632\) \([2]\) \(8847360\) \(3.1043\) \(\Gamma_0(N)\)-optimal
106722.gs3 106722gv2 \([1, -1, 1, -134310749, -597731686527]\) \(1763535241378513/4612311396\) \(700794201853268943603876\) \([2, 2]\) \(17694720\) \(3.4509\)  
106722.gs2 106722gv3 \([1, -1, 1, -187138139, -83763443739]\) \(4770223741048753/2740574865798\) \(416402712393107262353800038\) \([2]\) \(35389440\) \(3.7974\)  
106722.gs1 106722gv4 \([1, -1, 1, -2147621279, -38307037913427]\) \(7209828390823479793/49509306\) \(7522439749551389402586\) \([2]\) \(35389440\) \(3.7974\)  

Rank

sage: E.rank()
 

The elliptic curves in class 106722gv have rank \(1\).

Complex multiplication

The elliptic curves in class 106722gv do not have complex multiplication.

Modular form 106722.2.a.gv

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + 2 q^{5} + q^{8} + 2 q^{10} - 2 q^{13} + q^{16} + 2 q^{17} + O(q^{20})\) 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.