Properties

Label 103488ir
Number of curves $4$
Conductor $103488$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ir1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 103488ir

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
103488.ib3 103488ir1 \([0, 1, 0, -115046577, -475000133553]\) \(87364831012240243408/1760913\) \(3394267603550208\) \([2]\) \(8847360\) \(2.9635\) \(\Gamma_0(N)\)-optimal
103488.ib2 103488ir2 \([0, 1, 0, -115050497, -474966149505]\) \(21843440425782779332/3100814593569\) \(23908039794281629679616\) \([2, 2]\) \(17694720\) \(3.3101\)  
103488.ib4 103488ir3 \([0, 1, 0, -104678177, -564074750625]\) \(-8226100326647904626/4152140742401883\) \(-64028043667418530849357824\) \([2]\) \(35389440\) \(3.6567\)  
103488.ib1 103488ir4 \([0, 1, 0, -125485537, -383682506593]\) \(14171198121996897746/4077720290568771\) \(62880443854372908483084288\) \([2]\) \(35389440\) \(3.6567\)  

Rank

sage: E.rank()
 

The elliptic curves in class 103488ir have rank \(1\).

Complex multiplication

The elliptic curves in class 103488ir do not have complex multiplication.

Modular form 103488.2.a.ir

sage: E.q_eigenform(10)
 
\(q + q^{3} + 2q^{5} + q^{9} + q^{11} - 6q^{13} + 2q^{15} + 2q^{17} + 8q^{19} + O(q^{20})\)  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 Cremona 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 Cremona labels.