Properties

Label 1050.i
Number of curves $6$
Conductor $1050$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1050.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1050.i1 1050h4 \([1, 0, 1, -33601, 2367848]\) \(268498407453697/252\) \(3937500\) \([2]\) \(2048\) \(0.99382\)  
1050.i2 1050h5 \([1, 0, 1, -22851, -1318652]\) \(84448510979617/933897762\) \(14592152531250\) \([2]\) \(4096\) \(1.3404\)  
1050.i3 1050h3 \([1, 0, 1, -2601, 17848]\) \(124475734657/63011844\) \(984560062500\) \([2, 2]\) \(2048\) \(0.99382\)  
1050.i4 1050h2 \([1, 0, 1, -2101, 36848]\) \(65597103937/63504\) \(992250000\) \([2, 2]\) \(1024\) \(0.64725\)  
1050.i5 1050h1 \([1, 0, 1, -101, 848]\) \(-7189057/16128\) \(-252000000\) \([2]\) \(512\) \(0.30067\) \(\Gamma_0(N)\)-optimal
1050.i6 1050h6 \([1, 0, 1, 9649, 140348]\) \(6359387729183/4218578658\) \(-65915291531250\) \([2]\) \(4096\) \(1.3404\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1050.i have rank \(1\).

Complex multiplication

The elliptic curves in class 1050.i do not have complex multiplication.

Modular form 1050.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} - q^{8} + q^{9} - 4q^{11} + q^{12} - 6q^{13} - q^{14} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.