Properties

Label 433251.bh
Number of curves $2$
Conductor $433251$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("bh1")
 
E.isogeny_class()
 

Elliptic curves in class 433251.bh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
433251.bh1 433251bh2 \([1, -1, 0, -466677, -83646662]\) \(104154702625/32188247\) \(3473696489037509007\) \([2]\) \(6082560\) \(2.2622\) \(\Gamma_0(N)\)-optimal*
433251.bh2 433251bh1 \([1, -1, 0, 80838, -8856113]\) \(541343375/625807\) \(-67535941883231367\) \([2]\) \(3041280\) \(1.9156\) \(\Gamma_0(N)\)-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 433251.bh1.

Rank

sage: E.rank()
 

The elliptic curves in class 433251.bh have rank \(1\).

Complex multiplication

The elliptic curves in class 433251.bh do not have complex multiplication.

Modular form 433251.2.a.bh

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - q^{7} - 3 q^{8} - 4 q^{11} - q^{13} - q^{14} - q^{16} + 8 q^{17} - 4 q^{19} + O(q^{20})\) 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.