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Properties

Label 45570.cr
Number of curves $6$
Conductor $45570$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cr1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 45570.cr have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(7\)\(1\)
\(31\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 45570.cr do not have complex multiplication.

Modular form 45570.2.a.cr

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 45570.cr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
45570.cr1 45570da6 \([1, 0, 0, -601055561, 5671691591691]\) \(204117072508351537504018561/1958536333827640350\) \(230419841138488059537150\) \([2]\) \(15728640\) \(3.6446\)  
45570.cr2 45570da4 \([1, 0, 0, -129015531, -564034181355]\) \(2018651992700195476824481/75097045898437500\) \(8835092352905273437500\) \([2]\) \(7864320\) \(3.2980\)  
45570.cr3 45570da3 \([1, 0, 0, -38449811, 84228846141]\) \(53434051027037036566561/4869932476018522500\) \(572942685871103153602500\) \([2, 2]\) \(7864320\) \(3.2980\)  
45570.cr4 45570da2 \([1, 0, 0, -8437311, -7951546359]\) \(564608806599700366561/94616222006250000\) \(11131503902813306250000\) \([2, 2]\) \(3932160\) \(2.9514\)  
45570.cr5 45570da1 \([1, 0, 0, 974609, -702485575]\) \(870215264126076959/2316066449760000\) \(-272482901747814240000\) \([2]\) \(1966080\) \(2.6048\) \(\Gamma_0(N)\)-optimal
45570.cr6 45570da5 \([1, 0, 0, 43955939, 396431270591]\) \(79834049226144675685439/623263340520316720350\) \(-73326308748874741832457150\) \([2]\) \(15728640\) \(3.6446\)