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Properties

Label 8450.i
Number of curves $4$
Conductor $8450$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 8450.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8450.i1 8450g3 \([1, 1, 0, -876775, -303676875]\) \(988345570681/44994560\) \(3393439799360000000\) \([2]\) \(290304\) \(2.3176\)  
8450.i2 8450g1 \([1, 1, 0, -137400, 19430000]\) \(3803721481/26000\) \(1960891156250000\) \([2]\) \(96768\) \(1.7683\) \(\Gamma_0(N)\)-optimal
8450.i3 8450g2 \([1, 1, 0, -52900, 43174500]\) \(-217081801/10562500\) \(-796612032226562500\) \([2]\) \(193536\) \(2.1149\)  
8450.i4 8450g4 \([1, 1, 0, 475225, -1154084875]\) \(157376536199/7722894400\) \(-582452128062025000000\) \([2]\) \(580608\) \(2.6642\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8450.i have rank \(1\).

Complex multiplication

The elliptic curves in class 8450.i do not have complex multiplication.

Modular form 8450.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} - 4 q^{7} - q^{8} + q^{9} + 6 q^{11} + 2 q^{12} + 4 q^{14} + q^{16} + 6 q^{17} - q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.