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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gf1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 411840.2.a.gf

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

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 411840gf

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
411840.gf4 411840gf1 \([0, 0, 0, 508279092, -81738037574768]\) \(75991146714893572533071/15147028085515223040000\) \(-2894642144697541616247767040000\) \([2]\) \(619315200\) \(4.5243\) \(\Gamma_0(N)\)-optimal*
411840.gf3 411840gf2 \([0, 0, 0, -25443976908, -1517115793325168]\) \(9532597152396244075685450929/313550122650789880627200\) \(59920361563730954938542666547200\) \([2]\) \(1238630400\) \(4.8708\) \(\Gamma_0(N)\)-optimal*
411840.gf2 411840gf3 \([0, 0, 0, -127769593548, -17581781227125872]\) \(-1207087636168285491836819264689/236446260657750000000000\) \(-45185584075767742464000000000000\) \([2]\) \(1857945600\) \(5.0736\)  
411840.gf1 411840gf4 \([0, 0, 0, -2044409593548, -1125122938411125872]\) \(4944928228995290413834018379264689/189679641808585500000\) \(36248344036234711400448000000\) \([2]\) \(3715891200\) \(5.4202\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 411840gf1.