Properties

Label 784.b
Number of curves $6$
Conductor $784$
CM no
Rank $1$
Graph

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

Elliptic curves in class 784.b

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 \([0, 1, 0, 3512, -133260]\) \(9938375/21952\) \(-10578455953408\) \([2]\) \(1152\) \(1.1833\)  

Rank

sage: E.rank()
 

The elliptic curves in class 784.b have rank \(1\).

Complex multiplication

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

Modular form 784.2.a.b

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

Isogeny matrix

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

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

The vertices are labelled with LMFDB labels.