Properties

Label 189618.i
Number of curves $4$
Conductor $189618$
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 189618.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
189618.i1 189618bs4 \([1, 1, 0, -424131381339, -106316153461934307]\) \(1748094148784980747354970849498497/887694600425282263291392\) \(4284732286584156255995260528128\) \([2]\) \(1588543488\) \(5.2105\)  
189618.i2 189618bs3 \([1, 1, 0, -58018434139, 2958123786906397]\) \(4474676144192042711273397261697/1806328356954994499451382272\) \(8718801970305580044902427012930048\) \([2]\) \(1588543488\) \(5.2105\)  
189618.i3 189618bs2 \([1, 1, 0, -26651601499, -1642317815705315]\) \(433744050935826360922067531137/9612122270219882316693504\) \(46395878282997759945157055348736\) \([2, 2]\) \(794271744\) \(4.8639\)  
189618.i4 189618bs1 \([1, 1, 0, 151311781, -78662657863395]\) \(79374649975090937760383/553856914190911653543936\) \(-2673361538128920087530752180224\) \([2]\) \(397135872\) \(4.5173\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 189618.i have rank \(1\).

Complex multiplication

The elliptic curves in class 189618.i do not have complex multiplication.

Modular form 189618.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - q^{8} + q^{9} - 2q^{10} - q^{11} - q^{12} - 2q^{15} + q^{16} - q^{17} - q^{18} - 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.