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Properties

Label 12675e
Number of curves $8$
Conductor $12675$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("e1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 12675e have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(5\)\(1\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + 2 T + 2 T^{2}\) 1.2.c
\(7\) \( 1 + 3 T + 7 T^{2}\) 1.7.d
\(11\) \( 1 + 5 T + 11 T^{2}\) 1.11.f
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 + 6 T + 19 T^{2}\) 1.19.g
\(23\) \( 1 + 9 T + 23 T^{2}\) 1.23.j
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 12675e do not have complex multiplication.

Modular form 12675.2.a.e

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9} - 4 q^{11} + q^{12} - q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 12675e

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12675.j6 12675e1 \([1, 1, 1, -464838, 121785906]\) \(147281603041/5265\) \(397080459140625\) \([4]\) \(96768\) \(1.8903\) \(\Gamma_0(N)\)-optimal
12675.j5 12675e2 \([1, 1, 1, -485963, 110082656]\) \(168288035761/27720225\) \(2090628617375390625\) \([2, 2]\) \(193536\) \(2.2369\)  
12675.j4 12675e3 \([1, 1, 1, -2197088, -1149305344]\) \(15551989015681/1445900625\) \(109048221091494140625\) \([2, 2]\) \(387072\) \(2.5835\)  
12675.j7 12675e4 \([1, 1, 1, 887162, 620885156]\) \(1023887723039/2798036865\) \(-211024836286152890625\) \([2]\) \(387072\) \(2.5835\)  
12675.j2 12675e5 \([1, 1, 1, -34328213, -77428596094]\) \(59319456301170001/594140625\) \(44809426812744140625\) \([2, 2]\) \(774144\) \(2.9300\)  
12675.j8 12675e6 \([1, 1, 1, 2556037, -5436624094]\) \(24487529386319/183539412225\) \(-13842338855974062890625\) \([2]\) \(774144\) \(2.9300\)  
12675.j1 12675e7 \([1, 1, 1, -549250088, -4954768596094]\) \(242970740812818720001/24375\) \(1838335458984375\) \([2]\) \(1548288\) \(3.2766\)  
12675.j3 12675e8 \([1, 1, 1, -33504338, -81320581594]\) \(-55150149867714721/5950927734375\) \(-448812367916107177734375\) \([2]\) \(1548288\) \(3.2766\)