# Properties

 Label 3648.w Number of curves $4$ Conductor $3648$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("w1")

E.isogeny_class()

## Elliptic curves in class 3648.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3648.w1 3648bg3 $$[0, 1, 0, -5089, -137569]$$ $$111223479026/3518667$$ $$461198721024$$ $$[2]$$ $$3072$$ $$1.0128$$
3648.w2 3648bg2 $$[0, 1, 0, -769, 4991]$$ $$768400132/263169$$ $$17247043584$$ $$[2, 2]$$ $$1536$$ $$0.66620$$
3648.w3 3648bg1 $$[0, 1, 0, -689, 6735]$$ $$2211014608/513$$ $$8404992$$ $$[2]$$ $$768$$ $$0.31963$$ $$\Gamma_0(N)$$-optimal
3648.w4 3648bg4 $$[0, 1, 0, 2271, 37215]$$ $$9878111854/10097379$$ $$-1323483660288$$ $$[4]$$ $$3072$$ $$1.0128$$

## Rank

sage: E.rank()

The elliptic curves in class 3648.w have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3648.w do not have complex multiplication.

## Modular form3648.2.a.w

sage: E.q_eigenform(10)

$$q + q^{3} - 2 q^{5} + q^{9} - 2 q^{13} - 2 q^{15} + 2 q^{17} - q^{19} + O(q^{20})$$

## 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.