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Properties

Label 338.f
Number of curves $3$
Conductor $338$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 338.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338.f1 338c3 \([1, 0, 0, -77659, -8336303]\) \(-10730978619193/6656\) \(-32127240704\) \([]\) \(1008\) \(1.3369\)  
338.f2 338c2 \([1, 0, 0, -764, -16264]\) \(-10218313/17576\) \(-84835994984\) \([]\) \(336\) \(0.78755\)  
338.f3 338c1 \([1, 0, 0, 81, 467]\) \(12167/26\) \(-125497034\) \([]\) \(112\) \(0.23825\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 338.f have rank \(0\).

Complex multiplication

The elliptic curves in class 338.f do not have complex multiplication.

Modular form 338.2.a.f

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.