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Properties

Label 50430f
Number of curves $8$
Conductor $50430$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("f1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 50430f have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(41\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 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 50430f do not have complex multiplication.

Modular form 50430.2.a.f

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} + q^{10} - q^{12} - 2 q^{13} - 4 q^{14} + q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 50430f

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
50430.f8 50430f1 \([1, 1, 0, 2487, 147573]\) \(357911/2160\) \(-10260225160560\) \([2]\) \(138240\) \(1.1785\) \(\Gamma_0(N)\)-optimal
50430.f6 50430f2 \([1, 1, 0, -31133, 1902537]\) \(702595369/72900\) \(346282599168900\) \([2, 2]\) \(276480\) \(1.5251\)  
50430.f7 50430f3 \([1, 1, 0, -22728, -4325568]\) \(-273359449/1536000\) \(-7296160114176000\) \([2]\) \(414720\) \(1.7278\)  
50430.f5 50430f4 \([1, 1, 0, -115183, -13007933]\) \(35578826569/5314410\) \(25244001479412810\) \([2]\) \(552960\) \(1.8716\)  
50430.f4 50430f5 \([1, 1, 0, -485003, 129803103]\) \(2656166199049/33750\) \(160316018133750\) \([2]\) \(552960\) \(1.8716\)  
50430.f3 50430f6 \([1, 1, 0, -560648, -161505792]\) \(4102915888729/9000000\) \(42750938169000000\) \([2, 2]\) \(829440\) \(2.0744\)  
50430.f1 50430f7 \([1, 1, 0, -8965648, -10336598792]\) \(16778985534208729/81000\) \(384758443521000\) \([2]\) \(1658880\) \(2.4210\)  
50430.f2 50430f8 \([1, 1, 0, -762368, -35188728]\) \(10316097499609/5859375000\) \(27832642037109375000\) \([2]\) \(1658880\) \(2.4210\)