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Properties

Label 301665.bf
Number of curves $2$
Conductor $301665$
CM no
Rank $1$
Graph

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

Elliptic curves in class 301665.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
301665.bf1 301665bf2 \([0, 1, 1, -10144733915, -1020320251250569]\) \(-837552753022341363564544/2777578544133544921875\) \(-382912789086391589379780029296875\) \([]\) \(989853696\) \(4.9384\)  
301665.bf2 301665bf1 \([0, 1, 1, 1098589345, 32718724123934]\) \(1063644102334563516416/3970076690621356875\) \(-547308785093929221497507611875\) \([]\) \(329951232\) \(4.3891\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 301665.bf have rank \(1\).

Complex multiplication

The elliptic curves in class 301665.bf do not have complex multiplication.

Modular form 301665.2.a.bf

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{4} + q^{5} - q^{7} + q^{9} - 3 q^{11} - 2 q^{12} + q^{15} + 4 q^{16} - q^{17} - 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.