↑

Properties

Label 495495.bt
Number of curves $2$
Conductor $495495$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("bt1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 495495.bt have rank \(1\).

Complex multiplication

The elliptic curves in class 495495.bt do not have complex multiplication.

Modular form 495495.2.a.bt

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + q^{5} + q^{7} + 3 q^{8} - q^{10} + q^{13} - q^{14} - q^{16} - 4 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 495495.bt

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
495495.bt1 495495bt2 \([1, -1, 1, -143217317, -656136869066]\) \(9316717055063573427/57377784953125\) \(2000742427776539806734375\) \([2]\) \(80870400\) \(3.5013\) \(\Gamma_0(N)\)-optimal*
495495.bt2 495495bt1 \([1, -1, 1, -143004962, -658189832264]\) \(9275335480470938787/355047875\) \(12380389866648428625\) \([2]\) \(40435200\) \(3.1547\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 495495.bt1.