# Properties

 Label 38640.bx Number of curves $4$ Conductor $38640$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bx1")

sage: E.isogeny_class()

## Elliptic curves in class 38640.bx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.bx1 38640q4 $$[0, 1, 0, -131696, 7482804]$$ $$123343086124179938/59429226844575$$ $$121711056577689600$$ $$$$ $$393216$$ $$1.9721$$
38640.bx2 38640q2 $$[0, 1, 0, -108696, 13748004]$$ $$138697437757771876/106292300625$$ $$108843315840000$$ $$[2, 2]$$ $$196608$$ $$1.6255$$
38640.bx3 38640q1 $$[0, 1, 0, -108676, 13753340]$$ $$554483565352358224/326025$$ $$83462400$$ $$$$ $$98304$$ $$1.2789$$ $$\Gamma_0(N)$$-optimal
38640.bx4 38640q3 $$[0, 1, 0, -86016, 19672020]$$ $$-34366597532983298/61980408984375$$ $$-126935877600000000$$ $$$$ $$393216$$ $$1.9721$$

## Rank

sage: E.rank()

The elliptic curves in class 38640.bx have rank $$1$$.

## Complex multiplication

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

## Modular form 38640.2.a.bx

sage: E.q_eigenform(10)

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