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Properties

Label 57498b
Number of curves $6$
Conductor $57498$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("b1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 57498b have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(7\)\(1 + T\)
\(37\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 4 T + 23 T^{2}\) 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 57498b do not have complex multiplication.

Modular form 57498.2.a.b

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} - 6 q^{13} + q^{14} - 2 q^{15} + q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 57498b

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57498.f5 57498b1 \([1, 1, 0, -5504, 344832]\) \(-7189057/16128\) \(-41380035524352\) \([2]\) \(193536\) \(1.3014\) \(\Gamma_0(N)\)-optimal
57498.f4 57498b2 \([1, 1, 0, -115024, 14954800]\) \(65597103937/63504\) \(162933889877136\) \([2, 2]\) \(387072\) \(1.6480\)  
57498.f3 57498b3 \([1, 1, 0, -142404, 7261020]\) \(124475734657/63011844\) \(161671152230588196\) \([2, 2]\) \(774144\) \(1.9946\)  
57498.f1 57498b4 \([1, 1, 0, -1839964, 959876932]\) \(268498407453697/252\) \(646563055068\) \([2]\) \(774144\) \(1.9946\)  
57498.f6 57498b5 \([1, 1, 0, 528406, 56766798]\) \(6359387729183/4218578658\) \(-10823718671274379122\) \([2]\) \(1548288\) \(2.3411\)  
57498.f2 57498b6 \([1, 1, 0, -1251294, -534099078]\) \(84448510979617/933897762\) \(2396126151269396658\) \([2]\) \(1548288\) \(2.3411\)