# Properties

 Label 288.b Number of curves 4 Conductor 288 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("288.b1")

sage: E.isogeny_class()

## Elliptic curves in class 288.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
288.b1 288b2 [0, 0, 0, -291, -1910]  64
288.b2 288b3 [0, 0, 0, -156, 736]  64
288.b3 288b1 [0, 0, 0, -21, -20] [2, 2] 32 $$\Gamma_0(N)$$-optimal
288.b4 288b4 [0, 0, 0, 69, -146]  64

## Rank

sage: E.rank()

The elliptic curves in class 288.b have rank $$1$$.

## Modular form288.2.a.b

sage: E.q_eigenform(10)

$$q - 2q^{5} - 4q^{7} - 4q^{11} - 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 