# Properties

 Label 3864.d Number of curves $4$ Conductor $3864$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("d1")

E.isogeny_class()

## Elliptic curves in class 3864.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3864.d1 3864e3 $$[0, 1, 0, -18992, -1013472]$$ $$369937818893666/123409881$$ $$252743436288$$ $$$$ $$8192$$ $$1.1607$$
3864.d2 3864e4 $$[0, 1, 0, -9632, 352800]$$ $$48260105780546/1193313807$$ $$2443906676736$$ $$$$ $$8192$$ $$1.1607$$
3864.d3 3864e2 $$[0, 1, 0, -1352, -11520]$$ $$267100692772/102880449$$ $$105349579776$$ $$[2, 2]$$ $$4096$$ $$0.81410$$
3864.d4 3864e1 $$[0, 1, 0, 268, -1152]$$ $$8284506032/7394247$$ $$-1892927232$$ $$$$ $$2048$$ $$0.46752$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 3864.d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 3864.d do not have complex multiplication.

## Modular form3864.2.a.d

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 