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Properties

Label 34385.e
Number of curves $2$
Conductor $34385$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 34385.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
34385.e1 34385i1 \([1, 0, 0, -540, -1073]\) \(117649/65\) \(9622332785\) \([2]\) \(23760\) \(0.60613\) \(\Gamma_0(N)\)-optimal
34385.e2 34385i2 \([1, 0, 0, 2105, -7950]\) \(6967871/4225\) \(-625451631025\) \([2]\) \(47520\) \(0.95271\)  

Rank

sage: E.rank()
 

The elliptic curves in class 34385.e have rank \(1\).

Complex multiplication

The elliptic curves in class 34385.e do not have complex multiplication.

Modular form 34385.2.a.e

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