Properties

Label 411840.i
Number of curves $4$
Conductor $411840$
CM no
Rank $0$
Graph

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

Elliptic curves in class 411840.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
411840.i1 411840i4 \([0, 0, 0, -923790828, -10807085379152]\) \(912446049969377120252018/17177299425\) \(1641316519880294400\) \([2]\) \(88080384\) \(3.4832\)  
411840.i2 411840i3 \([0, 0, 0, -62886828, -136949964752]\) \(287849398425814280018/81784533026485575\) \(7814633826065840101785600\) \([2]\) \(88080384\) \(3.4832\) \(\Gamma_0(N)\)-optimal*
411840.i3 411840i2 \([0, 0, 0, -57738828, -168849031952]\) \(445574312599094932036/61129333175625\) \(2920499372689367040000\) \([2, 2]\) \(44040192\) \(3.1367\) \(\Gamma_0(N)\)-optimal*
411840.i4 411840i1 \([0, 0, 0, -3288828, -3125011952]\) \(-329381898333928144/162600887109375\) \(-1942094589177600000000\) \([2]\) \(22020096\) \(2.7901\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 411840.i1.

Rank

sage: E.rank()
 

The elliptic curves in class 411840.i have rank \(0\).

Complex multiplication

The elliptic curves in class 411840.i do not have complex multiplication.

Modular form 411840.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{5} - 4 q^{7} - q^{11} - q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.