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Properties

Label 45a
Number of curves $8$
Conductor $45$
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 45a have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(5\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 45a do not have complex multiplication.

Modular form 45.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 45a

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
45.a7 45a1 \([1, -1, 0, 0, -5]\) \(-1/15\) \(-10935\) \([2]\) \(2\) \(-0.54612\) \(\Gamma_0(N)\)-optimal
45.a6 45a2 \([1, -1, 0, -45, -104]\) \(13997521/225\) \(164025\) \([2, 2]\) \(4\) \(-0.19954\)  
45.a4 45a3 \([1, -1, 0, -720, -7259]\) \(56667352321/15\) \(10935\) \([2]\) \(8\) \(0.14703\)  
45.a5 45a4 \([1, -1, 0, -90, 175]\) \(111284641/50625\) \(36905625\) \([2, 2]\) \(8\) \(0.14703\)  
45.a2 45a5 \([1, -1, 0, -1215, 16600]\) \(272223782641/164025\) \(119574225\) \([2, 2]\) \(16\) \(0.49360\)  
45.a8 45a6 \([1, -1, 0, 315, 1066]\) \(4733169839/3515625\) \(-2562890625\) \([2]\) \(16\) \(0.49360\)  
45.a1 45a7 \([1, -1, 0, -19440, 1048135]\) \(1114544804970241/405\) \(295245\) \([2]\) \(32\) \(0.84018\)  
45.a3 45a8 \([1, -1, 0, -990, 22765]\) \(-147281603041/215233605\) \(-156905298045\) \([2]\) \(32\) \(0.84018\)