Properties

Label 8619.i
Number of curves $6$
Conductor $8619$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 8619.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8619.i1 8619b5 \([1, 1, 0, -3409409, -2424122748]\) \(908031902324522977/161726530797\) \(780623074389736773\) \([2]\) \(172032\) \(2.4376\)  
8619.i2 8619b3 \([1, 1, 0, -234744, -29790405]\) \(296380748763217/92608836489\) \(447005165444633601\) \([2, 2]\) \(86016\) \(2.0911\)  
8619.i3 8619b2 \([1, 1, 0, -91939, 10337800]\) \(17806161424897/668584449\) \(3227129435693241\) \([2, 2]\) \(43008\) \(1.7445\)  
8619.i4 8619b1 \([1, 1, 0, -91094, 10544487]\) \(17319700013617/25857\) \(124806800313\) \([2]\) \(21504\) \(1.3979\) \(\Gamma_0(N)\)-optimal
8619.i5 8619b4 \([1, 1, 0, 37346, 37254937]\) \(1193377118543/124806800313\) \(-602418587011991217\) \([2]\) \(86016\) \(2.0911\)  
8619.i6 8619b6 \([1, 1, 0, 655041, -200807082]\) \(6439735268725823/7345472585373\) \(-35455193184331664757\) \([4]\) \(172032\) \(2.4376\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8619.i have rank \(0\).

Complex multiplication

The elliptic curves in class 8619.i do not have complex multiplication.

Modular form 8619.2.a.i

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.