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Properties

Label 55770.d
Number of curves $6$
Conductor $55770$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 55770.d 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 \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 55770.2.a.d

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 55770.d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
55770.d1 55770a6 \([1, 1, 0, -28913368, -59852675228]\) \(553808571467029327441/12529687500\) \(60478408392187500\) \([2]\) \(2949120\) \(2.7442\)  
55770.d2 55770a4 \([1, 1, 0, -1998428, 1084165152]\) \(182864522286982801/463015182960\) \(2234885852247974640\) \([2]\) \(1474560\) \(2.3977\)  
55770.d3 55770a3 \([1, 1, 0, -1809148, -933521792]\) \(135670761487282321/643043610000\) \(3103848684140490000\) \([2, 2]\) \(1474560\) \(2.3977\)  
55770.d4 55770a5 \([1, 1, 0, -879648, -1890349092]\) \(-15595206456730321/310672490129100\) \(-1499556771407551041900\) \([2]\) \(2949120\) \(2.7442\)  
55770.d5 55770a2 \([1, 1, 0, -173228, 2551632]\) \(119102750067601/68309049600\) \(329714735390726400\) \([2, 2]\) \(737280\) \(2.0511\)  
55770.d6 55770a1 \([1, 1, 0, 43092, 345168]\) \(1833318007919/1070530560\) \(-5167246541783040\) \([2]\) \(368640\) \(1.7045\) \(\Gamma_0(N)\)-optimal