# Properties

 Label 67270y Number of curves $4$ Conductor $67270$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("67270.bd1")

sage: E.isogeny_class()

## Elliptic curves in class 67270y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
67270.bd4 67270y1 [1, -1, 1, 2222, 63681] [2] 115200 $$\Gamma_0(N)$$-optimal
67270.bd3 67270y2 [1, -1, 1, -16998, 694097] [2, 2] 230400
67270.bd2 67270y3 [1, -1, 1, -84268, -8777519] [2] 460800
67270.bd1 67270y4 [1, -1, 1, -257248, 50281697] [2] 460800

## Rank

sage: E.rank()

The elliptic curves in class 67270y have rank $$1$$.

## Modular form 67270.2.a.bd

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - 3q^{9} - q^{10} - 4q^{11} + 6q^{13} - q^{14} + q^{16} - 2q^{17} - 3q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.