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Properties

Label 55770cd
Number of curves $4$
Conductor $55770$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 55770cd have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(11\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 55770cd do not have complex multiplication.

Modular form 55770.2.a.cd

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

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 55770cd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
55770.ci4 55770cd1 \([1, 1, 1, -295677080, 716468940377]\) \(592265697637387401314569/296787655248366796800\) \(1432537325441714090095411200\) \([2]\) \(28385280\) \(3.9035\) \(\Gamma_0(N)\)-optimal
55770.ci2 55770cd2 \([1, 1, 1, -3839863960, 91510030757465]\) \(1297212465095901089487274249/1193746061037404160000\) \(5761984231129891736125440000\) \([2, 2]\) \(56770560\) \(4.2501\)  
55770.ci3 55770cd3 \([1, 1, 1, -2962470040, 134440564305497]\) \(-595697118196750093952139529/1272946549598037600000000\) \(-6144269862118754270018400000000\) \([2]\) \(113541120\) \(4.5967\)  
55770.ci1 55770cd4 \([1, 1, 1, -61424247960, 5859438337240665]\) \(5309860874757074224246393258249/4502770931800627200\) \(21734015258553653574604800\) \([2]\) \(113541120\) \(4.5967\)