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Properties

Label 333270k
Number of curves $2$
Conductor $333270$
CM no
Rank $1$
Graph

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

Elliptic curves in class 333270k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.k2 333270k1 \([1, -1, 0, -161145, -24988635]\) \(-1200031184926849/7291370520\) \(-2811851418703320\) \([]\) \(2322432\) \(1.8036\) \(\Gamma_0(N)\)-optimal
333270.k1 333270k2 \([1, -1, 0, -13071735, -18187371909]\) \(-640531120905493043809/141750\) \(-54664611750\) \([]\) \(6967296\) \(2.3529\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270k have rank \(1\).

Complex multiplication

The elliptic curves in class 333270k do not have complex multiplication.

Modular form 333270.2.a.k

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