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Properties

Label 35322w
Number of curves $2$
Conductor $35322$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 35322w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
35322.x2 35322w1 \([1, 1, 1, -2843859, 1121539329]\) \(6045996937/2204496\) \(927447412981359903696\) \([]\) \(2756160\) \(2.7236\) \(\Gamma_0(N)\)-optimal
35322.x1 35322w2 \([1, 1, 1, -98326794, -375272190441]\) \(249896037845497/37933056\) \(15958711040381589344256\) \([]\) \(8268480\) \(3.2730\)  

Rank

sage: E.rank()
 

The elliptic curves in class 35322w have rank \(1\).

Complex multiplication

The elliptic curves in class 35322w do not have complex multiplication.

Modular form 35322.2.a.w

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + 3 q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + 3 q^{10} - 6 q^{11} - q^{12} + 2 q^{13} + q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.