Properties

Label 12342bf
Number of curves $2$
Conductor $12342$
CM no
Rank $1$
Graph

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

Elliptic curves in class 12342bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12342.ba2 12342bf1 \([1, 0, 0, -66129, -6410007]\) \(18052771191337/444958272\) \(788270721302592\) \([2]\) \(80640\) \(1.6421\) \(\Gamma_0(N)\)-optimal
12342.ba1 12342bf2 \([1, 0, 0, -148409, 12695409]\) \(204055591784617/78708537864\) \(139436976046885704\) \([2]\) \(161280\) \(1.9887\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12342bf have rank \(1\).

Complex multiplication

The elliptic curves in class 12342bf do not have complex multiplication.

Modular form 12342.2.a.bf

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