Properties

Label 2730.w
Number of curves $8$
Conductor $2730$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 2730.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2730.w1 2730v7 \([1, 1, 1, -9218020, -10769952415]\) \(86623684689189325642735681/56690726941459561860\) \(56690726941459561860\) \([2]\) \(131072\) \(2.7300\)  
2730.w2 2730v4 \([1, 1, 1, -5591040, 5086138017]\) \(19328649688935739391016961/1048320\) \(1048320\) \([4]\) \(32768\) \(2.0368\)  
2730.w3 2730v5 \([1, 1, 1, -688720, -98092255]\) \(36128658497509929012481/16775330746084419600\) \(16775330746084419600\) \([2, 2]\) \(65536\) \(2.3834\)  
2730.w4 2730v3 \([1, 1, 1, -350720, 78749345]\) \(4770955732122964500481/71987251059360000\) \(71987251059360000\) \([2, 4]\) \(32768\) \(2.0368\)  
2730.w5 2730v2 \([1, 1, 1, -349440, 79361697]\) \(4718909406724749250561/1098974822400\) \(1098974822400\) \([2, 4]\) \(16384\) \(1.6903\)  
2730.w6 2730v6 \([1, 1, 1, -33200, 216426017]\) \(-4047051964543660801/20235220197806250000\) \(-20235220197806250000\) \([8]\) \(65536\) \(2.3834\)  
2730.w7 2730v1 \([1, 1, 1, -21760, 1242785]\) \(-1139466686381936641/17587891077120\) \(-17587891077120\) \([4]\) \(8192\) \(1.3437\) \(\Gamma_0(N)\)-optimal
2730.w8 2730v8 \([1, 1, 1, 2432580, -737334495]\) \(1591934139020114746758719/1156766383092650262660\) \(-1156766383092650262660\) \([2]\) \(131072\) \(2.7300\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2730.w have rank \(0\).

Complex multiplication

The elliptic curves in class 2730.w do not have complex multiplication.

Modular form 2730.2.a.w

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

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 8 & 16 & 4 \\ 16 & 1 & 8 & 4 & 2 & 8 & 4 & 16 \\ 2 & 8 & 1 & 2 & 4 & 4 & 8 & 2 \\ 4 & 4 & 2 & 1 & 2 & 2 & 4 & 4 \\ 8 & 2 & 4 & 2 & 1 & 4 & 2 & 8 \\ 8 & 8 & 4 & 2 & 4 & 1 & 8 & 8 \\ 16 & 4 & 8 & 4 & 2 & 8 & 1 & 16 \\ 4 & 16 & 2 & 4 & 8 & 8 & 16 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.