Properties

Label 409101bh
Number of curves $2$
Conductor $409101$
CM no
Rank $1$
Graph

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

Elliptic curves in class 409101bh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
409101.bh2 409101bh1 \([1, 1, 0, -58566, 9742671]\) \(-36561310759/46662561\) \(-28354291617108303\) \([2]\) \(2764800\) \(1.8502\) \(\Gamma_0(N)\)-optimal*
409101.bh1 409101bh2 \([1, 1, 0, -1130021, 461682390]\) \(262623524091319/134454573\) \(81700705884869379\) \([2]\) \(5529600\) \(2.1967\) \(\Gamma_0(N)\)-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 409101bh1.

Rank

sage: E.rank()
 

The elliptic curves in class 409101bh have rank \(1\).

Complex multiplication

The elliptic curves in class 409101bh do not have complex multiplication.

Modular form 409101.2.a.bh

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} - 2 q^{5} - q^{6} - 3 q^{8} + q^{9} - 2 q^{10} + q^{12} + 2 q^{13} + 2 q^{15} - q^{16} + q^{18} + 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.