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Properties

Label 336973k
Number of curves $2$
Conductor $336973$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 336973k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
336973.k1 336973k1 \([0, -1, 1, -1132419367, 14667977243632]\) \(-9221261135586623488/121324931\) \(-2113028278326466282691\) \([]\) \(87588864\) \(3.6502\) \(\Gamma_0(N)\)-optimal
336973.k2 336973k2 \([0, -1, 1, -1068394497, 16399573996245]\) \(-7743965038771437568/2189290237869371\) \(-38129279315906281774535249531\) \([]\) \(262766592\) \(4.1996\)  

Rank

sage: E.rank()
 

The elliptic curves in class 336973k have rank \(1\).

Complex multiplication

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

Modular form 336973.2.a.k

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