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Properties

Label 453882bf
Number of curves $4$
Conductor $453882$
CM no
Rank $1$
Graph

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 453882bf have rank \(1\).

Complex multiplication

The elliptic curves in class 453882bf do not have complex multiplication.

Modular form 453882.2.a.bf

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} + 4 q^{7} - q^{8} + q^{9} + q^{11} + q^{12} + q^{13} - 4 q^{14} + q^{16} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 453882bf

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
453882.bf4 453882bf1 \([1, 0, 1, 190164, -22934678]\) \(5137417856375/4510142208\) \(-667662911277702912\) \([2]\) \(7299072\) \(2.1077\) \(\Gamma_0(N)\)-optimal*
453882.bf3 453882bf2 \([1, 0, 1, -952476, -203928854]\) \(645532578015625/252306960048\) \(37350485131593162672\) \([2]\) \(14598144\) \(2.4543\) \(\Gamma_0(N)\)-optimal*
453882.bf2 453882bf3 \([1, 0, 1, -1976091, 1427589670]\) \(-5764706497797625/2612665516032\) \(-386768262325440872448\) \([2]\) \(21897216\) \(2.6570\) \(\Gamma_0(N)\)-optimal*
453882.bf1 453882bf4 \([1, 0, 1, -34477851, 77910731302]\) \(30618029936661765625/3678951124992\) \(544616800375740837888\) \([2]\) \(43794432\) \(3.0036\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 453882bf1.