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Properties

Label 49686.bj
Number of curves $2$
Conductor $49686$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("bj1")
 
E.isogeny_class()
 

Elliptic curves in class 49686.bj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
49686.bj1 49686bg2 \([1, 0, 1, -23833, 1467260]\) \(-6329617441/279936\) \(-66208682606976\) \([]\) \(197568\) \(1.4170\)  
49686.bj2 49686bg1 \([1, 0, 1, -173, -2026]\) \(-2401/6\) \(-1419081846\) \([]\) \(28224\) \(0.44400\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 49686.bj have rank \(1\).

Complex multiplication

The elliptic curves in class 49686.bj do not have complex multiplication.

Modular form 49686.2.a.bj

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} - 5 q^{11} + q^{12} + q^{15} + q^{16} + 4 q^{17} - q^{18} + 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.