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Properties

Label 448448.bt
Number of curves $2$
Conductor $448448$
CM no
Rank $1$
Graph

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 448448.bt have rank \(1\).

Complex multiplication

The elliptic curves in class 448448.bt do not have complex multiplication.

Modular form 448448.2.a.bt

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + 3 q^{5} - 2 q^{9} + q^{11} + q^{13} - 3 q^{15} - 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 448448.bt

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
448448.bt1 448448bt2 \([0, -1, 0, -327189, -73552499]\) \(-2009615368192/53094899\) \(-102343668079837184\) \([]\) \(4147200\) \(2.0464\)  
448448.bt2 448448bt1 \([0, -1, 0, 17771, -420979]\) \(321978368/224939\) \(-433583692365824\) \([]\) \(1382400\) \(1.4971\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 448448.bt1.