Properties

Label 333270.cy
Number of curves $2$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cy1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333270.cy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.cy1 333270cy1 \([1, -1, 1, -2606172293, 21411682137981]\) \(489781415227546051766883/233890092903563264000\) \(934851451444332634457505792000\) \([2]\) \(544997376\) \(4.4447\) \(\Gamma_0(N)\)-optimal
333270.cy2 333270cy2 \([1, -1, 1, 9354475387, 162848733083517]\) \(22649115256119592694355357/15973509811739648000000\) \(-63845623586639738562130944000000\) \([2]\) \(1089994752\) \(4.7913\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270.cy have rank \(1\).

Complex multiplication

The elliptic curves in class 333270.cy do not have complex multiplication.

Modular form 333270.2.a.cy

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} - 2q^{11} - 6q^{13} - q^{14} + q^{16} + 6q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.