Properties

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

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Show commands: SageMath
E = EllipticCurve("bg1")
 
E.isogeny_class()
 

Elliptic curves in class 333270.bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.bg1 333270bg1 \([1, -1, 0, -2710695, -1713396739]\) \(551105805571803/1376829440\) \(5503144594097848320\) \([2]\) \(14192640\) \(2.4734\) \(\Gamma_0(N)\)-optimal
333270.bg2 333270bg2 \([1, -1, 0, -1695015, -3013670275]\) \(-134745327251163/903920796800\) \(-3612943405976661590400\) \([2]\) \(28385280\) \(2.8199\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 333270.2.a.bg

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