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Properties

Label 59150.cd
Number of curves $6$
Conductor $59150$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 59150.cd have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 59150.cd do not have complex multiplication.

Modular form 59150.2.a.cd

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{7} + q^{8} + q^{9} + 2 q^{12} + q^{14} + q^{16} + q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 6 & 18 \\ 2 & 1 & 6 & 18 & 3 & 9 \\ 3 & 6 & 1 & 3 & 2 & 6 \\ 9 & 18 & 3 & 1 & 6 & 2 \\ 6 & 3 & 2 & 6 & 1 & 3 \\ 18 & 9 & 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 59150.cd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
59150.cd1 59150bx6 \([1, 1, 1, -2235219438, -40656632425219]\) \(16375858190544687071329/9025573730468750\) \(680698758006095886230468750\) \([2]\) \(41803776\) \(4.0967\)  
59150.cd2 59150bx5 \([1, 1, 1, -2234923688, -40667933624219]\) \(16369358802802724130049/4976562500\) \(375326822875976562500\) \([2]\) \(20901888\) \(3.7502\)  
59150.cd3 59150bx4 \([1, 1, 1, -86004188, 241034475781]\) \(932829715460155969/206949435875000\) \(15607896869162076171875000\) \([2]\) \(13934592\) \(3.5474\)  
59150.cd4 59150bx2 \([1, 1, 1, -80765188, 279339339781]\) \(772531501373731009/15142400\) \(1142023009400000000\) \([2]\) \(4644864\) \(2.9981\)  
59150.cd5 59150bx3 \([1, 1, 1, -28037188, -53901620219]\) \(32318182904349889/2067798824000\) \(155951093341759625000000\) \([2]\) \(6967296\) \(3.2009\)  
59150.cd6 59150bx1 \([1, 1, 1, -5053188, 4353355781]\) \(189208196468929/834928640\) \(62969391779840000000\) \([2]\) \(2322432\) \(2.6516\) \(\Gamma_0(N)\)-optimal