Properties

Label 285318.x
Number of curves $2$
Conductor $285318$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 285318.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
285318.x1 285318x1 \([1, -1, 1, -77342, -8218515]\) \(39616946929/226368\) \(292347021206592\) \([2]\) \(3317760\) \(1.6172\) \(\Gamma_0(N)\)-optimal
285318.x2 285318x2 \([1, -1, 1, -33782, -17453235]\) \(-3301293169/100082952\) \(-129253926750964488\) \([2]\) \(6635520\) \(1.9638\)  

Rank

sage: E.rank()
 

The elliptic curves in class 285318.x have rank \(2\).

Complex multiplication

The elliptic curves in class 285318.x do not have complex multiplication.

Modular form 285318.2.a.x

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - 4 q^{5} + 4 q^{7} + q^{8} - 4 q^{10} - 6 q^{13} + 4 q^{14} + q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 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.