↑

Properties

Label 27075g
Number of curves $8$
Conductor $27075$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("g1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 27075g have rank \(2\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 5 T + 11 T^{2}\) 1.11.af
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(23\) \( 1 - 9 T + 23 T^{2}\) 1.23.aj
\(29\) \( 1 + 7 T + 29 T^{2}\) 1.29.h
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 27075g do not have complex multiplication.

Modular form 27075.2.a.g

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9} - 4 q^{11} + q^{12} - 2 q^{13} - q^{16} - 2 q^{17} - q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 27075g

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27075.f7 27075g1 \([1, 1, 1, -188, -159844]\) \(-1/15\) \(-11026378359375\) \([2]\) \(41472\) \(1.1815\) \(\Gamma_0(N)\)-optimal
27075.f6 27075g2 \([1, 1, 1, -45313, -3679594]\) \(13997521/225\) \(165395675390625\) \([2, 2]\) \(82944\) \(1.5281\)  
27075.f5 27075g3 \([1, 1, 1, -90438, 4803906]\) \(111284641/50625\) \(37214026962890625\) \([2, 2]\) \(165888\) \(1.8747\)  
27075.f4 27075g4 \([1, 1, 1, -722188, -236524594]\) \(56667352321/15\) \(11026378359375\) \([2]\) \(165888\) \(1.8747\)  
27075.f8 27075g5 \([1, 1, 1, 315687, 36481656]\) \(4733169839/3515625\) \(-2584307427978515625\) \([2]\) \(331776\) \(2.2212\)  
27075.f2 27075g6 \([1, 1, 1, -1218563, 516972656]\) \(272223782641/164025\) \(120573447359765625\) \([2, 2]\) \(331776\) \(2.2212\)  
27075.f3 27075g7 \([1, 1, 1, -992938, 714620156]\) \(-147281603041/215233605\) \(-158216477625484453125\) \([2]\) \(663552\) \(2.5678\)  
27075.f1 27075g8 \([1, 1, 1, -19494188, 33120687656]\) \(1114544804970241/405\) \(297712215703125\) \([2]\) \(663552\) \(2.5678\)