Properties

Label 6800.b
Number of curves $4$
Conductor $6800$
CM no
Rank $1$
Graph

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

Elliptic curves in class 6800.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6800.b1 6800t4 \([0, 1, 0, -45208, 2541588]\) \(159661140625/48275138\) \(3089608832000000\) \([2]\) \(41472\) \(1.6774\)  
6800.b2 6800t3 \([0, 1, 0, -41208, 3205588]\) \(120920208625/19652\) \(1257728000000\) \([2]\) \(20736\) \(1.3308\)  
6800.b3 6800t2 \([0, 1, 0, -17208, -874412]\) \(8805624625/2312\) \(147968000000\) \([2]\) \(13824\) \(1.1280\)  
6800.b4 6800t1 \([0, 1, 0, -1208, -10412]\) \(3048625/1088\) \(69632000000\) \([2]\) \(6912\) \(0.78147\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 6800.b have rank \(1\).

Complex multiplication

The elliptic curves in class 6800.b do not have complex multiplication.

Modular form 6800.2.a.b

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} - 4 q^{7} + q^{9} - 6 q^{11} - 2 q^{13} + q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.