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Properties

Label 784.b
Number of curves $6$
Conductor $784$
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 784.b have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 784.b do not have complex multiplication.

Modular form 784.2.a.b

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 784.b

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
784.b1 784j6 \([0, 1, 0, -2140728, -1206278060]\) \(2251439055699625/25088\) \(12089663946752\) \([2]\) \(6912\) \(2.0792\)  
784.b2 784j5 \([0, 1, 0, -133688, -18913196]\) \(-548347731625/1835008\) \(-884272562962432\) \([2]\) \(3456\) \(1.7326\)  
784.b3 784j4 \([0, 1, 0, -27848, -1475468]\) \(4956477625/941192\) \(453551299002368\) \([2]\) \(2304\) \(1.5299\)  
784.b4 784j2 \([0, 1, 0, -8248, 285396]\) \(128787625/98\) \(47225249792\) \([2]\) \(768\) \(0.98059\)  
784.b5 784j1 \([0, 1, 0, -408, 6292]\) \(-15625/28\) \(-13492928512\) \([2]\) \(384\) \(0.63402\) \(\Gamma_0(N)\)-optimal
784.b6 784j3 \([0, 1, 0, 3512, -133260]\) \(9938375/21952\) \(-10578455953408\) \([2]\) \(1152\) \(1.1833\)