Properties

Label 474320.hd
Number of curves $2$
Conductor $474320$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("hd1")
 
E.isogeny_class()
 

Elliptic curves in class 474320.hd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
474320.hd1 474320hd1 \([0, 1, 0, -334000, 74338900]\) \(-584043889/1400\) \(-9877498317209600\) \([]\) \(3981312\) \(1.9485\) \(\Gamma_0(N)\)-optimal
474320.hd2 474320hd2 \([0, 1, 0, 614640, 375247508]\) \(3639707951/10718750\) \(-75624596491136000000\) \([]\) \(11943936\) \(2.4978\)  

Rank

sage: E.rank()
 

The elliptic curves in class 474320.hd have rank \(1\).

Complex multiplication

The elliptic curves in class 474320.hd do not have complex multiplication.

Modular form 474320.2.a.hd

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} - 2 q^{9} - 5 q^{13} + q^{15} - 6 q^{17} + 5 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.