Properties

Label 10320.k
Number of curves $4$
Conductor $10320$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 10320.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10320.k1 10320v4 \([0, -1, 0, -82920, 7957872]\) \(15393836938735081/2275690697640\) \(9321229097533440\) \([2]\) \(69120\) \(1.7889\)  
10320.k2 10320v3 \([0, -1, 0, -79720, 8690032]\) \(13679527032530281/381633600\) \(1563171225600\) \([2]\) \(34560\) \(1.4423\)  
10320.k3 10320v2 \([0, -1, 0, -21720, -1223568]\) \(276670733768281/336980250\) \(1380271104000\) \([2]\) \(23040\) \(1.2396\)  
10320.k4 10320v1 \([0, -1, 0, -1720, -7568]\) \(137467988281/72562500\) \(297216000000\) \([2]\) \(11520\) \(0.89304\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 10320.k have rank \(1\).

Complex multiplication

The elliptic curves in class 10320.k do not have complex multiplication.

Modular form 10320.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} - 2 q^{7} + q^{9} + 6 q^{11} + 2 q^{13} - q^{15} - 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.