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Properties

Label 479808pi
Number of curves $2$
Conductor $479808$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("pi1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 479808pi have rank \(1\).

Complex multiplication

The elliptic curves in class 479808pi do not have complex multiplication.

Modular form 479808.2.a.pi

Copy content sage:E.q_eigenform(10)
 
\(q + 3 q^{5} - 6 q^{11} - 5 q^{13} + q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 479808pi

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
479808.pi2 479808pi1 \([0, 0, 0, -7510675116, -250528205022928]\) \(42531320912955257257/1127938881456\) \(1242617134917992489058041856\) \([]\) \(650280960\) \(4.3042\) \(\Gamma_0(N)\)-optimal*
479808.pi1 479808pi2 \([0, 0, 0, -13031712876, 163901533122992]\) \(222165413800219579417/118033833938006016\) \(130034407859139534620514809020416\) \([]\) \(1950842880\) \(4.8535\) \(\Gamma_0(N)\)-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 479808pi1.