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Properties

Label 15a
Number of curves $8$
Conductor $15$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("a1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 15a have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1 - T\)
 
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
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 15a do not have complex multiplication.

Modular form 15.2.a.a

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3 q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{15} - q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 15a

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
15.a5 15a1 \([1, 1, 1, -10, -10]\) \(111284641/50625\) \(50625\) \([2, 4]\) \(1\) \(-0.40228\) \(\Gamma_0(N)\)-optimal
15.a2 15a2 \([1, 1, 1, -135, -660]\) \(272223782641/164025\) \(164025\) \([2, 2]\) \(2\) \(-0.055704\)  
15.a6 15a3 \([1, 1, 1, -5, 2]\) \(13997521/225\) \(225\) \([2, 4]\) \(2\) \(-0.74885\)  
15.a8 15a4 \([1, 1, 1, 35, -28]\) \(4733169839/3515625\) \(-3515625\) \([8]\) \(2\) \(-0.055704\)  
15.a1 15a5 \([1, 1, 1, -2160, -39540]\) \(1114544804970241/405\) \(405\) \([2]\) \(4\) \(0.29087\)  
15.a3 15a6 \([1, 1, 1, -110, -880]\) \(-147281603041/215233605\) \(-215233605\) \([2]\) \(4\) \(0.29087\)  
15.a4 15a7 \([1, 1, 1, -80, 242]\) \(56667352321/15\) \(15\) \([4]\) \(4\) \(-0.40228\)  
15.a7 15a8 \([1, 1, 1, 0, 0]\) \(-1/15\) \(-15\) \([4]\) \(4\) \(-1.0954\)