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Properties

Label 199410.bc
Number of curves $2$
Conductor $199410$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("bc1")
 
E.isogeny_class()
 

Elliptic curves in class 199410.bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
199410.bc1 199410bh2 \([1, 0, 1, -217768, -17356642]\) \(47316161414809/22001657400\) \(531066523606860600\) \([2]\) \(3440640\) \(2.0957\)  
199410.bc2 199410bh1 \([1, 0, 1, 48112, -2041954]\) \(510273943271/370215360\) \(-8936098796859840\) \([2]\) \(1720320\) \(1.7491\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 199410.bc have rank \(1\).

Complex multiplication

The elliptic curves in class 199410.bc do not have complex multiplication.

Modular form 199410.2.a.bc

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} - q^{10} + 2 q^{11} + q^{12} + 4 q^{14} + q^{15} + q^{16} - q^{18} + O(q^{20})\) 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.