Properties

Label 115710.bl
Number of curves $8$
Conductor $115710$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bl1")
 
E.isogeny_class()
 

Elliptic curves in class 115710.bl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
115710.bl1 115710bk8 \([1, 0, 1, -16061619848, -783488938704514]\) \(458238674512589140162512851063637241/392468692344120\) \(392468692344120\) \([2]\) \(79626240\) \(3.9829\)  
115710.bl2 115710bk7 \([1, 0, 1, -1005154648, -12208693503874]\) \(112311018900728619777866774440441/605137023657856954297300680\) \(605137023657856954297300680\) \([2]\) \(79626240\) \(3.9829\)  
115710.bl3 115710bk6 \([1, 0, 1, -1003851248, -12242077227394]\) \(111874678752008055543640205694841/3274073546848947662400\) \(3274073546848947662400\) \([2, 2]\) \(39813120\) \(3.6363\)  
115710.bl4 115710bk5 \([1, 0, 1, -198296123, -1074708019744]\) \(862315869068763192029111676841/81885458731960991976750\) \(81885458731960991976750\) \([6]\) \(26542080\) \(3.4336\)  
115710.bl5 115710bk4 \([1, 0, 1, -73328623, 229914901256]\) \(43605803425346349400425156841/2397733750298041341023250\) \(2397733750298041341023250\) \([6]\) \(26542080\) \(3.4336\)  
115710.bl6 115710bk3 \([1, 0, 1, -62659248, -191807812994]\) \(-27206911093874365984986366841/147776899215941936640000\) \(-147776899215941936640000\) \([2]\) \(19906560\) \(3.2898\)  
115710.bl7 115710bk2 \([1, 0, 1, -13312373, -14159184244]\) \(260909623523179393528416841/64394497051541618062500\) \(64394497051541618062500\) \([2, 6]\) \(13271040\) \(3.0870\)  
115710.bl8 115710bk1 \([1, 0, 1, 2000127, -1400809244]\) \(884905188895571476583159/1359998795777343750000\) \(-1359998795777343750000\) \([6]\) \(6635520\) \(2.7405\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 115710.bl have rank \(1\).

Complex multiplication

The elliptic curves in class 115710.bl do not have complex multiplication.

Modular form 115710.2.a.bl

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.