# Properties

 Label 227430fz Number of curves $4$ Conductor $227430$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("227430.z1")

sage: E.isogeny_class()

## Elliptic curves in class 227430fz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
227430.z4 227430fz1 [1, -1, 0, 9760470, 5089608820] [2] 22394880 $$\Gamma_0(N)$$-optimal
227430.z3 227430fz2 [1, -1, 0, -42916650, 42585182836] [2] 44789760
227430.z2 227430fz3 [1, -1, 0, -177035370, 928235782196] [2] 67184640
227430.z1 227430fz4 [1, -1, 0, -2851222290, 58600285736300] [2] 134369280

## Rank

sage: E.rank()

The elliptic curves in class 227430fz have rank $$1$$.

## Modular form 227430.2.a.z

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} + q^{10} - 2q^{13} - q^{14} + q^{16} - 6q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.