Properties

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

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

Elliptic curves in class 50430.f

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

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 50430.f do not have complex multiplication.

Modular form 50430.2.a.f

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.