Properties

Label 816.b
Number of curves $6$
Conductor $816$
CM no
Rank $1$
Graph

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

Elliptic curves in class 816.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
816.b1 816h5 \([0, -1, 0, -443904, 113984640]\) \(2361739090258884097/5202\) \(21307392\) \([4]\) \(3072\) \(1.5402\)  
816.b2 816h3 \([0, -1, 0, -27744, 1787904]\) \(576615941610337/27060804\) \(110841053184\) \([2, 4]\) \(1536\) \(1.1937\)  
816.b3 816h6 \([0, -1, 0, -26304, 1980288]\) \(-491411892194497/125563633938\) \(-514308644610048\) \([4]\) \(3072\) \(1.5402\)  
816.b4 816h2 \([0, -1, 0, -1824, 25344]\) \(163936758817/30338064\) \(124264710144\) \([2, 2]\) \(768\) \(0.84708\)  
816.b5 816h1 \([0, -1, 0, -544, -4352]\) \(4354703137/352512\) \(1443889152\) \([2]\) \(384\) \(0.50050\) \(\Gamma_0(N)\)-optimal
816.b6 816h4 \([0, -1, 0, 3616, 142848]\) \(1276229915423/2927177028\) \(-11989717106688\) \([2]\) \(1536\) \(1.1937\)  

Rank

sage: E.rank()
 

The elliptic curves in class 816.b have rank \(1\).

Complex multiplication

The elliptic curves in class 816.b do not have complex multiplication.

Modular form 816.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + q^{17} - 4 q^{19} + O(q^{20})\) Copy content 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.