# Properties

 Label 90.b Number of curves 4 Conductor 90 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("90.b1")

sage: E.isogeny_class()

## Elliptic curves in class 90.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
90.b1 90b4 [1, -1, 1, -218, -269]  48
90.b2 90b2 [1, -1, 1, -128, 587]  16
90.b3 90b1 [1, -1, 1, -8, 11]  8 $$\Gamma_0(N)$$-optimal
90.b4 90b3 [1, -1, 1, 52, -53]  24

## Rank

sage: E.rank()

The elliptic curves in class 90.b have rank $$0$$.

## Modular form90.2.a.b

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} + 2q^{7} + q^{8} - q^{10} - 6q^{11} - 4q^{13} + 2q^{14} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 