Properties

Label 28830h
Number of curves $8$
Conductor $28830$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 28830h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28830.a8 28830h1 \([1, 1, 0, 1422, -62748]\) \(357911/2160\) \(-1917007950960\) \([2]\) \(57600\) \(1.0387\) \(\Gamma_0(N)\)-optimal
28830.a6 28830h2 \([1, 1, 0, -17798, -835392]\) \(702595369/72900\) \(64699018344900\) \([2, 2]\) \(115200\) \(1.3853\)  
28830.a7 28830h3 \([1, 1, 0, -12993, 1860213]\) \(-273359449/1536000\) \(-1363205654016000\) \([2]\) \(172800\) \(1.5880\)  
28830.a5 28830h4 \([1, 1, 0, -65848, 5574478]\) \(35578826569/5314410\) \(4716558437343210\) \([2]\) \(230400\) \(1.7319\)  
28830.a4 28830h5 \([1, 1, 0, -277268, -56310078]\) \(2656166199049/33750\) \(29953249233750\) \([2]\) \(230400\) \(1.7319\)  
28830.a3 28830h6 \([1, 1, 0, -320513, 69576117]\) \(4102915888729/9000000\) \(7987533129000000\) \([2, 2]\) \(345600\) \(1.9346\)  
28830.a1 28830h7 \([1, 1, 0, -5125513, 4464229117]\) \(16778985534208729/81000\) \(71887798161000\) \([2]\) \(691200\) \(2.2812\)  
28830.a2 28830h8 \([1, 1, 0, -435833, 14891373]\) \(10316097499609/5859375000\) \(5200216880859375000\) \([2]\) \(691200\) \(2.2812\)  

Rank

sage: E.rank()
 

The elliptic curves in class 28830h have rank \(2\).

Complex multiplication

The elliptic curves in class 28830h do not have complex multiplication.

Modular form 28830.2.a.h

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} - q^{12} - 2 q^{13} + 4 q^{14} + q^{15} + q^{16} - 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.