↑

Properties

Label 6930ba
Number of curves $8$
Conductor $6930$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("ba1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 6930ba have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 6930ba do not have complex multiplication.

Modular form 6930.2.a.ba

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} + q^{11} + 2 q^{13} + q^{14} + q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 6930ba

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6930.z7 6930ba1 \([1, -1, 1, -10553, -187063]\) \(178272935636041/81841914000\) \(59662755306000\) \([2]\) \(18432\) \(1.3381\) \(\Gamma_0(N)\)-optimal
6930.z5 6930ba2 \([1, -1, 1, -141773, -20499919]\) \(432288716775559561/270140062500\) \(196932105562500\) \([2, 2]\) \(36864\) \(1.6847\)  
6930.z4 6930ba3 \([1, -1, 1, -429728, 108528707]\) \(12038605770121350841/757333463040\) \(552096094556160\) \([6]\) \(55296\) \(1.8874\)  
6930.z2 6930ba4 \([1, -1, 1, -2268023, -1314110419]\) \(1769857772964702379561/691787250\) \(504312905250\) \([2]\) \(73728\) \(2.0313\)  
6930.z6 6930ba5 \([1, -1, 1, -115043, -28486843]\) \(-230979395175477481/348191894531250\) \(-253831891113281250\) \([2]\) \(73728\) \(2.0313\)  
6930.z3 6930ba6 \([1, -1, 1, -455648, 94718531]\) \(14351050585434661561/3001282273281600\) \(2187934777222286400\) \([2, 6]\) \(110592\) \(2.2340\)  
6930.z1 6930ba7 \([1, -1, 1, -2307848, -1265537149]\) \(1864737106103260904761/129177711985836360\) \(94170552037674706440\) \([6]\) \(221184\) \(2.5806\)  
6930.z8 6930ba8 \([1, -1, 1, 981832, 570811907]\) \(143584693754978072519/276341298967965000\) \(-201452806947646485000\) \([6]\) \(221184\) \(2.5806\)