Properties

Label 127050ga
Number of curves $8$
Conductor $127050$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ga1")
 
E.isogeny_class()
 

Elliptic curves in class 127050ga

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
127050.gv8 127050ga1 \([1, 1, 1, 4806662, -505870969]\) \(443688652450511/260789760000\) \(-7218827625240000000000\) \([2]\) \(9953280\) \(2.8836\) \(\Gamma_0(N)\)-optimal
127050.gv7 127050ga2 \([1, 1, 1, -19393338, -4087470969]\) \(29141055407581489/16604321025600\) \(459618243131765025000000\) \([2, 2]\) \(19906560\) \(3.2302\)  
127050.gv6 127050ga3 \([1, 1, 1, -61259338, 202591177031]\) \(-918468938249433649/109183593750000\) \(-3022271820739746093750000\) \([2]\) \(29859840\) \(3.4329\)  
127050.gv5 127050ga4 \([1, 1, 1, -199078338, 1076178749031]\) \(31522423139920199089/164434491947880\) \(4551652077963722510625000\) \([2]\) \(39813120\) \(3.5768\)  
127050.gv4 127050ga5 \([1, 1, 1, -226908338, -1313092090969]\) \(46676570542430835889/106752955783320\) \(2954990204694596289375000\) \([2]\) \(39813120\) \(3.5768\)  
127050.gv3 127050ga6 \([1, 1, 1, -1006571838, 12291247427031]\) \(4074571110566294433649/48828650062500\) \(1351608314583946289062500\) \([2, 2]\) \(59719680\) \(3.7795\)  
127050.gv1 127050ga7 \([1, 1, 1, -16105103088, 786664718177031]\) \(16689299266861680229173649/2396798250\) \(66344911008878906250\) \([2]\) \(119439360\) \(4.1261\)  
127050.gv2 127050ga8 \([1, 1, 1, -1033040588, 11610682927031]\) \(4404531606962679693649/444872222400201750\) \(12314348112305059569246093750\) \([2]\) \(119439360\) \(4.1261\)  

Rank

sage: E.rank()
 

The elliptic curves in class 127050ga have rank \(0\).

Complex multiplication

The elliptic curves in class 127050ga do not have complex multiplication.

Modular form 127050.2.a.ga

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.