Properties

Label 188370.ca
Number of curves $8$
Conductor $188370$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ca1")
 
E.isogeny_class()
 

Elliptic curves in class 188370.ca

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
188370.ca1 188370u7 \([1, -1, 1, -4488970883, 115763459349251]\) \(13722604572968640187892492722921/36939806611960382108160\) \(26929119020119118556848640\) \([6]\) \(159252480\) \(4.1139\)  
188370.ca2 188370u8 \([1, -1, 1, -797732483, -6404131470589]\) \(77013704252633562960444236521/20262661472595628847255040\) \(14771480213522213429648924160\) \([6]\) \(159252480\) \(4.1139\)  
188370.ca3 188370u5 \([1, -1, 1, -738264308, -7720679532769]\) \(61042428203425827148268361721/2287149206968899000\) \(1667331771880327371000\) \([2]\) \(53084160\) \(3.5646\)  
188370.ca4 188370u6 \([1, -1, 1, -284055683, 1761480412931]\) \(3477015524751011858387583721/173605868128473455001600\) \(126558677865657148696166400\) \([2, 6]\) \(79626240\) \(3.7674\)  
188370.ca5 188370u4 \([1, -1, 1, -74154308, 42318131231]\) \(61859347930211625693801721/34737934177406743101000\) \(25323954015329515720629000\) \([2]\) \(53084160\) \(3.5646\)  
188370.ca6 188370u2 \([1, -1, 1, -46209308, -120254700769]\) \(14968716721822395621081721/91209357028881000000\) \(66491621274054249000000\) \([2, 2]\) \(26542080\) \(3.2181\)  
188370.ca7 188370u1 \([1, -1, 1, -1209308, -4046700769]\) \(-268291321601301081721/9550359000000000000\) \(-6962211711000000000000\) \([2]\) \(13271040\) \(2.8715\) \(\Gamma_0(N)\)-optimal
188370.ca8 188370u3 \([1, -1, 1, 10856317, 107731881731]\) \(194108149567956675968279/6990401110687088640000\) \(-5096002409690887618560000\) \([6]\) \(39813120\) \(3.4208\)  

Rank

sage: E.rank()
 

The elliptic curves in class 188370.ca have rank \(0\).

Complex multiplication

The elliptic curves in class 188370.ca do not have complex multiplication.

Modular form 188370.2.a.ca

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

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.