Properties

Label 6930.d
Number of curves $6$
Conductor $6930$
CM no
Rank $0$
Graph

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

Elliptic curves in class 6930.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6930.d1 6930i3 \([1, -1, 0, -802898370, -8756469168620]\) \(78519570041710065450485106721/96428056919040\) \(70296053493980160\) \([2]\) \(1474560\) \(3.4067\)  
6930.d2 6930i5 \([1, -1, 0, -236147490, 1278495746356]\) \(1997773216431678333214187041/187585177195046990066400\) \(136749594175189255758405600\) \([2]\) \(2949120\) \(3.7533\)  
6930.d3 6930i4 \([1, -1, 0, -52439490, -123820900844]\) \(21876183941534093095979041/3572502915711058560000\) \(2604354625553361690240000\) \([2, 2]\) \(1474560\) \(3.4067\)  
6930.d4 6930i2 \([1, -1, 0, -50181570, -136808005100]\) \(19170300594578891358373921/671785075055001600\) \(489731319715096166400\) \([2, 2]\) \(737280\) \(3.0602\)  
6930.d5 6930i1 \([1, -1, 0, -2995650, -2337570284]\) \(-4078208988807294650401/880065599546327040\) \(-641567822069272412160\) \([2]\) \(368640\) \(2.7136\) \(\Gamma_0(N)\)-optimal
6930.d6 6930i6 \([1, -1, 0, 95141790, -694989970700]\) \(130650216943167617311657439/361816948816603087500000\) \(-263764555687303650787500000\) \([2]\) \(2949120\) \(3.7533\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6930.d have rank \(0\).

Complex multiplication

The elliptic curves in class 6930.d do not have complex multiplication.

Modular form 6930.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} + q^{10} + q^{11} - 2 q^{13} - q^{14} + q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.