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Properties

Label 411840.if
Number of curves $2$
Conductor $411840$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("if1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 411840.if have rank \(1\).

Complex multiplication

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

Modular form 411840.2.a.if

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 411840.if

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
411840.if1 411840if2 \([0, 0, 0, -180237612, -276682436656]\) \(3388383326345613179401/1787816842064922240\) \(341657119061528625272586240\) \([2]\) \(115605504\) \(3.7829\) \(\Gamma_0(N)\)-optimal*
411840.if2 411840if1 \([0, 0, 0, 42789588, -33761210416]\) \(45338857965533777399/28814396538470400\) \(-5506516930145791927910400\) \([2]\) \(57802752\) \(3.4363\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 411840.if1.