Properties

Label 259350.gs
Number of curves $4$
Conductor $259350$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("gs1")
 
E.isogeny_class()
 

Elliptic curves in class 259350.gs

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259350.gs1 259350gs3 \([1, 0, 0, -2713213, 1719887417]\) \(141369383441705190409/6345626621880\) \(99150415966875000\) \([2]\) \(5898240\) \(2.3380\)  
259350.gs2 259350gs4 \([1, 0, 0, -843213, -275942583]\) \(4243415895694547209/351514682293320\) \(5492416910833125000\) \([2]\) \(5898240\) \(2.3380\)  
259350.gs3 259350gs2 \([1, 0, 0, -178213, 23972417]\) \(40061018056412809/7275103617600\) \(113673494025000000\) \([2, 2]\) \(2949120\) \(1.9915\)  
259350.gs4 259350gs1 \([1, 0, 0, 21787, 2172417]\) \(73197245859191/172623360000\) \(-2697240000000000\) \([2]\) \(1474560\) \(1.6449\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 259350.gs have rank \(0\).

Complex multiplication

The elliptic curves in class 259350.gs do not have complex multiplication.

Modular form 259350.2.a.gs

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.