# Properties

 Label 1290b Number of curves $2$ Conductor $1290$ CM no Rank $0$ Graph

# Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 1290b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1290.b1 1290b1 $$[1, 1, 0, -154527, -23386059]$$ $$408076159454905367161/1190206406250000$$ $$1190206406250000$$ $$[2]$$ $$10560$$ $$1.7632$$ $$\Gamma_0(N)$$-optimal
1290.b2 1290b2 $$[1, 1, 0, -92027, -42398559]$$ $$-86193969101536367161/725294740213012500$$ $$-725294740213012500$$ $$[2]$$ $$21120$$ $$2.1098$$

## Rank

sage: E.rank()

The elliptic curves in class 1290b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 1290b do not have complex multiplication.

## Modular form1290.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} + 6q^{11} - q^{12} + 2q^{13} - q^{15} + q^{16} - q^{18} - 2q^{19} + O(q^{20})$$

## 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 Cremona 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 Cremona labels.