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Properties

Label 58800gb
Number of curves $6$
Conductor $58800$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 58800gb have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 7 T + 19 T^{2}\) 1.19.h
\(23\) \( 1 + 5 T + 23 T^{2}\) 1.23.f
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 58800gb do not have complex multiplication.

Modular form 58800.2.a.gb

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{9} - 4 q^{11} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 58800gb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
58800.m6 58800gb1 \([0, -1, 0, 19192, -231888]\) \(103823/63\) \(-474360768000000\) \([2]\) \(196608\) \(1.5053\) \(\Gamma_0(N)\)-optimal
58800.m5 58800gb2 \([0, -1, 0, -78808, -1799888]\) \(7189057/3969\) \(29884728384000000\) \([2, 2]\) \(393216\) \(1.8519\)  
58800.m3 58800gb3 \([0, -1, 0, -764808, 256136112]\) \(6570725617/45927\) \(345808999872000000\) \([2]\) \(786432\) \(2.1985\)  
58800.m2 58800gb4 \([0, -1, 0, -960808, -361655888]\) \(13027640977/21609\) \(162705743424000000\) \([2, 2]\) \(786432\) \(2.1985\)  
58800.m4 58800gb5 \([0, -1, 0, -666808, -587447888]\) \(-4354703137/17294403\) \(-130218829987008000000\) \([2]\) \(1572864\) \(2.5450\)  
58800.m1 58800gb6 \([0, -1, 0, -15366808, -23180759888]\) \(53297461115137/147\) \(1106841792000000\) \([2]\) \(1572864\) \(2.5450\)