Properties

Label 316050.il
Number of curves $4$
Conductor $316050$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("il1")
 
E.isogeny_class()
 

Elliptic curves in class 316050.il

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
316050.il1 316050il4 \([1, 0, 0, -1027188, -398056758]\) \(65202655558249/512820150\) \(942699653552343750\) \([2]\) \(5308416\) \(2.2772\)  
316050.il2 316050il2 \([1, 0, 0, -108438, 3436992]\) \(76711450249/41602500\) \(76476445664062500\) \([2, 2]\) \(2654208\) \(1.9306\)  
316050.il3 316050il1 \([1, 0, 0, -83938, 9341492]\) \(35578826569/51600\) \(94854506250000\) \([2]\) \(1327104\) \(1.5840\) \(\Gamma_0(N)\)-optimal
316050.il4 316050il3 \([1, 0, 0, 418312, 27140742]\) \(4403686064471/2721093750\) \(-5002093103027343750\) \([2]\) \(5308416\) \(2.2772\)  

Rank

sage: E.rank()
 

The elliptic curves in class 316050.il have rank \(1\).

Complex multiplication

The elliptic curves in class 316050.il do not have complex multiplication.

Modular form 316050.2.a.il

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.