Properties

Label 250050.bt
Number of curves $2$
Conductor $250050$
CM no
Rank $1$
Graph

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

Elliptic curves in class 250050.bt

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
250050.bt1 250050bt1 \([1, 0, 0, -2538, -49308]\) \(115714886617/320064\) \(5001000000\) \([2]\) \(196608\) \(0.73391\) \(\Gamma_0(N)\)-optimal
250050.bt2 250050bt2 \([1, 0, 0, -1538, -88308]\) \(-25750777177/200080008\) \(-3126250125000\) \([2]\) \(393216\) \(1.0805\)  

Rank

sage: E.rank()
 

The elliptic curves in class 250050.bt have rank \(1\).

Complex multiplication

The elliptic curves in class 250050.bt do not have complex multiplication.

Modular form 250050.2.a.bt

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} + q^{8} + q^{9} + q^{12} + 2 q^{13} + 2 q^{14} + q^{16} - 6 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.