↑

Properties

Label 1470k
Number of curves $8$
Conductor $1470$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 1470k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1470.j7 1470k1 \([1, 1, 1, 10289, -298411]\) \(1023887723039/928972800\) \(-109292720947200\) \([4]\) \(6144\) \(1.3811\) \(\Gamma_0(N)\)-optimal
1470.j6 1470k2 \([1, 1, 1, -52431, -2731947]\) \(135487869158881/51438240000\) \(6051657497760000\) \([2, 2]\) \(12288\) \(1.7277\)  
1470.j4 1470k3 \([1, 1, 1, -738431, -244478347]\) \(378499465220294881/120530818800\) \(14180330301001200\) \([2]\) \(24576\) \(2.0742\)  
1470.j5 1470k4 \([1, 1, 1, -369951, 84522549]\) \(47595748626367201/1215506250000\) \(143003094806250000\) \([2, 2]\) \(24576\) \(2.0742\)  
1470.j2 1470k5 \([1, 1, 1, -5882451, 5488977549]\) \(191342053882402567201/129708022500\) \(15260019139102500\) \([2, 2]\) \(49152\) \(2.4208\)  
1470.j8 1470k6 \([1, 1, 1, 62229, 270705693]\) \(226523624554079/269165039062500\) \(-31666997680664062500\) \([2]\) \(49152\) \(2.4208\)  
1470.j1 1470k7 \([1, 1, 1, -94119201, 351412332249]\) \(783736670177727068275201/360150\) \(42371287350\) \([2]\) \(98304\) \(2.7674\)  
1470.j3 1470k8 \([1, 1, 1, -5845701, 5560992849]\) \(-187778242790732059201/4984939585440150\) \(-586473157287448207350\) \([2]\) \(98304\) \(2.7674\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1470k have rank \(1\).

Complex multiplication

The elliptic curves in class 1470k do not have complex multiplication.

Modular form 1470.2.a.k

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - 4 q^{11} - q^{12} + 2 q^{13} + q^{15} + q^{16} - 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.