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Properties

Label 76230ej
Number of curves $8$
Conductor $76230$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ej1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 76230ej have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1 + T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 76230ej do not have complex multiplication.

Modular form 76230.2.a.ej

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

Isogeny matrix

Copy content 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 76230ej

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
76230.ee8 76230ej1 \([1, -1, 1, 1730398, 109268129]\) \(443688652450511/260789760000\) \(-336801621683197440000\) \([4]\) \(3317760\) \(2.6282\) \(\Gamma_0(N)\)-optimal
76230.ee7 76230ej2 \([1, -1, 1, -6981602, 882893729]\) \(29141055407581489/16604321025600\) \(21443948751555629006400\) \([2, 2]\) \(6635520\) \(2.9748\)  
76230.ee6 76230ej3 \([1, -1, 1, -22053362, -43759694239]\) \(-918468938249433649/109183593750000\) \(-141007114068433593750000\) \([4]\) \(9953280\) \(3.1775\)  
76230.ee5 76230ej4 \([1, -1, 1, -71668202, -232454609791]\) \(31522423139920199089/164434491947880\) \(212361879349475437455720\) \([2]\) \(13271040\) \(3.3214\)  
76230.ee4 76230ej5 \([1, -1, 1, -81687002, 283627891649]\) \(46676570542430835889/106752955783320\) \(137868022990231084477080\) \([2]\) \(13271040\) \(3.3214\)  
76230.ee3 76230ej6 \([1, -1, 1, -362365862, -2654909444239]\) \(4074571110566294433649/48828650062500\) \(63060637525228598062500\) \([2, 2]\) \(19906560\) \(3.5241\)  
76230.ee1 76230ej7 \([1, -1, 1, -5797837112, -169919579126239]\) \(16689299266861680229173649/2396798250\) \(3095388168030254250\) \([2]\) \(39813120\) \(3.8707\)  
76230.ee2 76230ej8 \([1, -1, 1, -371894612, -2507907512239]\) \(4404531606962679693649/444872222400201750\) \(574538225527704859262745750\) \([2]\) \(39813120\) \(3.8707\)