# Properties

 Label 3648.x Number of curves $4$ Conductor $3648$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 3648.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3648.x1 3648r3 $$[0, 1, 0, -5603329, -5107121569]$$ $$74220219816682217473/16416$$ $$4303355904$$ $$$$ $$46080$$ $$2.1412$$
3648.x2 3648r2 $$[0, 1, 0, -350209, -79885729]$$ $$18120364883707393/269485056$$ $$70643890520064$$ $$[2, 2]$$ $$23040$$ $$1.7946$$
3648.x3 3648r4 $$[0, 1, 0, -339969, -84766113]$$ $$-16576888679672833/2216253521952$$ $$-580977563258585088$$ $$$$ $$46080$$ $$2.1412$$
3648.x4 3648r1 $$[0, 1, 0, -22529, -1176993]$$ $$4824238966273/537919488$$ $$141012366262272$$ $$$$ $$11520$$ $$1.4480$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form3648.2.a.x

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} + q^{9} + 4q^{11} - 2q^{13} - 2q^{15} - 6q^{17} + q^{19} + O(q^{20})$$ ## 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. 