# Properties

 Label 588.c Number of curves $4$ Conductor $588$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("588.c1")

sage: E.isogeny_class()

## Elliptic curves in class 588.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
588.c1 588b4 [0, -1, 0, -89588, 10350936]  1728
588.c2 588b3 [0, -1, 0, -5553, 165894]  864
588.c3 588b2 [0, -1, 0, -1388, 6840]  576
588.c4 588b1 [0, -1, 0, 327, 666]  288 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form588.2.a.c

sage: E.q_eigenform(10)

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

## 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 LMFDB 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 LMFDB labels. 