Properties

Label 6699d
Number of curves $4$
Conductor $6699$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 6699d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6699.a4 6699d1 \([1, 1, 1, -33814, 2404610]\) \(-4275768267198290017/52843101620463\) \(-52843101620463\) \([4]\) \(20736\) \(1.4439\) \(\Gamma_0(N)\)-optimal
6699.a3 6699d2 \([1, 1, 1, -542619, 153621456]\) \(17668869054438249282097/2649918412449\) \(2649918412449\) \([2, 2]\) \(41472\) \(1.7905\)  
6699.a2 6699d3 \([1, 1, 1, -544214, 152670836]\) \(17825137625614555960417/216318148151991039\) \(216318148151991039\) \([2]\) \(82944\) \(2.1370\)  
6699.a1 6699d4 \([1, 1, 1, -8681904, 9842626320]\) \(72371679832051361738355457/1627857\) \(1627857\) \([2]\) \(82944\) \(2.1370\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6699d have rank \(1\).

Complex multiplication

The elliptic curves in class 6699d do not have complex multiplication.

Modular form 6699.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} - q^{7} + 3 q^{8} + q^{9} + 2 q^{10} + q^{11} + q^{12} + 2 q^{13} + q^{14} + 2 q^{15} - q^{16} + 2 q^{17} - q^{18} - 8 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.