# Properties

 Label 463680.kj Number of curves $4$ Conductor $463680$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("kj1")

sage: E.isogeny_class()

## Elliptic curves in class 463680.kj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.kj1 463680kj4 $$[0, 0, 0, -6677292, -6641232176]$$ $$344577854816148242/2716875$$ $$259601448960000$$ $$$$ $$7864320$$ $$2.3568$$
463680.kj2 463680kj2 $$[0, 0, 0, -417612, -103622384]$$ $$168591300897604/472410225$$ $$22569749972582400$$ $$[2, 2]$$ $$3932160$$ $$2.0102$$
463680.kj3 463680kj3 $$[0, 0, 0, -252012, -186621104]$$ $$-18524646126002/146738831715$$ $$-14021113717749841920$$ $$$$ $$7864320$$ $$2.3568$$
463680.kj4 463680kj1 $$[0, 0, 0, -36732, -175376]$$ $$458891455696/264449745$$ $$3158570829496320$$ $$$$ $$1966080$$ $$1.6636$$ $$\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 463680.kj1.

## Rank

sage: E.rank()

The elliptic curves in class 463680.kj have rank $$0$$.

## Complex multiplication

The elliptic curves in class 463680.kj do not have complex multiplication.

## Modular form 463680.2.a.kj

sage: E.q_eigenform(10)

$$q + q^{5} - q^{7} + 4 q^{11} - 2 q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 