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Properties

Label 336973.e
Number of curves $2$
Conductor $336973$
CM no
Rank $0$
Graph

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

Elliptic curves in class 336973.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
336973.e1 336973e2 \([1, 0, 0, -2540798, -1064316275]\) \(104154702625/32188247\) \(560599339147837993367\) \([2]\) \(12165120\) \(2.6859\)  
336973.e2 336973e1 \([1, 0, 0, 440117, -112212024]\) \(541343375/625807\) \(-10899226373964728527\) \([2]\) \(6082560\) \(2.3393\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 336973.e have rank \(0\).

Complex multiplication

The elliptic curves in class 336973.e do not have complex multiplication.

Modular form 336973.2.a.e

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