# Properties

 Label 42432s Number of curves $6$ Conductor $42432$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("42432.cf1")

sage: E.isogeny_class()

## Elliptic curves in class 42432s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
42432.cf4 42432s1 [0, 1, 0, -34497, 2454687] [2] 65536 $$\Gamma_0(N)$$-optimal
42432.cf3 42432s2 [0, 1, 0, -34817, 2406495] [2, 2] 131072
42432.cf5 42432s3 [0, 1, 0, 14143, 8683167] [2] 262144
42432.cf2 42432s4 [0, 1, 0, -88897, -6949345] [2, 2] 262144
42432.cf6 42432s5 [0, 1, 0, 248063, -46778017] [2] 524288
42432.cf1 42432s6 [0, 1, 0, -1291137, -565029153] [2] 524288

## Rank

sage: E.rank()

The elliptic curves in class 42432s have rank $$1$$.

## Modular form 42432.2.a.cf

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.