Properties

Label 80724j
Number of curves $4$
Conductor $80724$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 80724j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
80724.h4 80724j1 \([0, -1, 0, 6407, 65170]\) \(2048000/1323\) \(-18786677919408\) \([2]\) \(172800\) \(1.2359\) \(\Gamma_0(N)\)-optimal
80724.h3 80724j2 \([0, -1, 0, -27228, 562968]\) \(9826000/5103\) \(1159406408740608\) \([2]\) \(345600\) \(1.5825\)  
80724.h2 80724j3 \([0, -1, 0, -108913, 14284126]\) \(-10061824000/352947\) \(-5011868187166512\) \([2]\) \(518400\) \(1.7852\)  
80724.h1 80724j4 \([0, -1, 0, -1757028, 897014520]\) \(2640279346000/3087\) \(701369308991232\) \([2]\) \(1036800\) \(2.1318\)  

Rank

sage: E.rank()
 

The elliptic curves in class 80724j have rank \(0\).

Complex multiplication

The elliptic curves in class 80724j do not have complex multiplication.

Modular form 80724.2.a.j

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{7} + q^{9} + 6 q^{11} - 2 q^{13} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.