↑

Properties

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

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("ek1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 76230ek 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
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 76230.2.a.ek

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 76230ek

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
76230.ea7 76230ek1 \([1, -1, 1, -44672, -1267549]\) \(7633736209/3870720\) \(4998910896967680\) \([2]\) \(552960\) \(1.7048\) \(\Gamma_0(N)\)-optimal
76230.ea5 76230ek2 \([1, -1, 1, -393152, 94076579]\) \(5203798902289/57153600\) \(73812043713038400\) \([2, 2]\) \(1105920\) \(2.0514\)  
76230.ea4 76230ek3 \([1, -1, 1, -2919632, -1919440861]\) \(2131200347946769/2058000\) \(2657841080202000\) \([2]\) \(1658880\) \(2.2541\)  
76230.ea6 76230ek4 \([1, -1, 1, -88232, 236047331]\) \(-58818484369/18600435000\) \(-24021866011966515000\) \([2]\) \(2211840\) \(2.3980\)  
76230.ea2 76230ek5 \([1, -1, 1, -6273752, 6049948259]\) \(21145699168383889/2593080\) \(3348879761054520\) \([2]\) \(2211840\) \(2.3980\)  
76230.ea3 76230ek6 \([1, -1, 1, -2941412, -1889332189]\) \(2179252305146449/66177562500\) \(85466202235245562500\) \([2, 2]\) \(3317760\) \(2.6007\)  
76230.ea8 76230ek7 \([1, -1, 1, 793858, -6362691541]\) \(42841933504271/13565917968750\) \(-17519948526722167968750\) \([2]\) \(6635520\) \(2.9473\)  
76230.ea1 76230ek8 \([1, -1, 1, -7025162, 4510720811]\) \(29689921233686449/10380965400750\) \(13406684302365873576750\) \([2]\) \(6635520\) \(2.9473\)