Properties

Label 323400bf
Number of curves $4$
Conductor $323400$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bf1")
 
E.isogeny_class()
 

Elliptic curves in class 323400bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
323400.bf3 323400bf1 \([0, -1, 0, -289508, -28940988]\) \(5702413264/2525985\) \(1188718437060000000\) \([2]\) \(4718592\) \(2.1640\) \(\Gamma_0(N)\)-optimal
323400.bf2 323400bf2 \([0, -1, 0, -2274008, 1300674012]\) \(690862540036/12006225\) \(22600325840400000000\) \([2, 2]\) \(9437184\) \(2.5105\)  
323400.bf1 323400bf3 \([0, -1, 0, -36231008, 83952012012]\) \(1397097631688978/433125\) \(1630615140000000000\) \([2]\) \(18874368\) \(2.8571\)  
323400.bf4 323400bf4 \([0, -1, 0, -69008, 3712944012]\) \(-9653618/1581886845\) \(-5955436973676960000000\) \([2]\) \(18874368\) \(2.8571\)  

Rank

sage: E.rank()
 

The elliptic curves in class 323400bf have rank \(1\).

Complex multiplication

The elliptic curves in class 323400bf do not have complex multiplication.

Modular form 323400.2.a.bf

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.