# Properties

 Label 38640.j Number of curves $4$ Conductor $38640$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 38640.j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.j1 38640b4 $$[0, -1, 0, -220816, -39865184]$$ $$581416486276209698/12425175$$ $$25446758400$$ $$$$ $$131072$$ $$1.5244$$
38640.j2 38640b2 $$[0, -1, 0, -13816, -617984]$$ $$284840777767396/1312250625$$ $$1343744640000$$ $$[2, 2]$$ $$65536$$ $$1.1778$$
38640.j3 38640b3 $$[0, -1, 0, -6816, -1250784]$$ $$-17101973157698/321306440175$$ $$-658035589478400$$ $$$$ $$131072$$ $$1.5244$$
38640.j4 38640b1 $$[0, -1, 0, -1316, 2016]$$ $$985329269584/566015625$$ $$144900000000$$ $$$$ $$32768$$ $$0.83124$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 38640.j have rank $$0$$.

## Complex multiplication

The elliptic curves in class 38640.j do not have complex multiplication.

## Modular form 38640.2.a.j

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} - q^{7} + q^{9} + 2q^{13} + q^{15} + 6q^{17} - 4q^{19} + O(q^{20})$$ ## 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. 