Properties

Label 296450gc
Number of curves $4$
Conductor $296450$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("gc1")
 
E.isogeny_class()
 

Elliptic curves in class 296450gc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
296450.gc4 296450gc1 \([1, 0, 0, 1479162, 5659296292]\) \(109902239/4312000\) \(-14042457858496375000000\) \([2]\) \(26542080\) \(2.9296\) \(\Gamma_0(N)\)-optimal
296450.gc2 296450gc2 \([1, 0, 0, -40023838, 93272129292]\) \(2177286259681/105875000\) \(344792492061294921875000\) \([2]\) \(53084160\) \(3.2761\)  
296450.gc3 296450gc3 \([1, 0, 0, -13343338, -154794266208]\) \(-80677568161/3131816380\) \(-10199072248770563403437500\) \([2]\) \(79626240\) \(3.4789\)  
296450.gc1 296450gc4 \([1, 0, 0, -521755088, -4562215726958]\) \(4823468134087681/30382271150\) \(98942894774880611442968750\) \([2]\) \(159252480\) \(3.8254\)  

Rank

sage: E.rank()
 

The elliptic curves in class 296450gc have rank \(2\).

Complex multiplication

The elliptic curves in class 296450gc do not have complex multiplication.

Modular form 296450.2.a.gc

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.