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Properties

Label 75810cc
Number of curves $4$
Conductor $75810$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cc1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 75810cc have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(7\)\(1 + T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 75810cc do not have complex multiplication.

Modular form 75810.2.a.cc

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + 2 q^{13} - q^{14} + q^{15} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 75810cc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75810.cc4 75810cc1 \([1, 1, 1, 1249594, -5554540861]\) \(4586790226340951/286015269335040\) \(-13455840325319240970240\) \([2]\) \(5806080\) \(2.9258\) \(\Gamma_0(N)\)-optimal
75810.cc3 75810cc2 \([1, 1, 1, -40453126, -95332156477]\) \(155617476551393929129/6633105589454400\) \(312060296221906557326400\) \([2, 2]\) \(11612160\) \(3.2723\)  
75810.cc2 75810cc3 \([1, 1, 1, -107671326, 303513755043]\) \(2934284984699764805929/851931751022747640\) \(40079879778737813764470840\) \([2]\) \(23224320\) \(3.6189\)  
75810.cc1 75810cc4 \([1, 1, 1, -640478446, -6239111413021]\) \(617611911727813844500009/1197723879765000\) \(56347975118282497965000\) \([2]\) \(23224320\) \(3.6189\)