# Properties

 Label 588.b Number of curves $2$ Conductor $588$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("588.b1")

sage: E.isogeny_class()

## Elliptic curves in class 588.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
588.b1 588c2 [0, -1, 0, -44, 120]  96
588.b2 588c1 [0, -1, 0, -9, -6]  48 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 588.b have rank $$1$$.

## Modular form588.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{5} + q^{9} + 2q^{11} + 4q^{13} + 2q^{15} - 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 