# Properties

 Label 4032z Number of curves 6 Conductor 4032 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("4032.r1")

sage: E.isogeny_class()

## Elliptic curves in class 4032z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4032.r5 4032z1 [0, 0, 0, -300, 4048] [2] 1536 $$\Gamma_0(N)$$-optimal
4032.r4 4032z2 [0, 0, 0, -6060, 181456] [2] 3072
4032.r6 4032z3 [0, 0, 0, 2580, -84656] [2] 4608
4032.r3 4032z4 [0, 0, 0, -20460, -923312] [2] 9216
4032.r2 4032z5 [0, 0, 0, -98220, -11882288] [2] 13824
4032.r1 4032z6 [0, 0, 0, -1572780, -759189296] [2] 27648

## Rank

sage: E.rank()

The elliptic curves in class 4032z have rank $$1$$.

## Modular form4032.2.a.r

sage: E.q_eigenform(10)

$$q - q^{7} + 4q^{13} - 6q^{17} + 2q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.