# Properties

 Label 1290i Number of curves $2$ Conductor $1290$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 1290i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1290.i2 1290i1 $$[1, 0, 1, 2, -4]$$ $$1685159/7740$$ $$-7740$$ $$$$ $$112$$ $$-0.57142$$ $$\Gamma_0(N)$$-optimal
1290.i1 1290i2 $$[1, 0, 1, -28, -52]$$ $$2305199161/277350$$ $$277350$$ $$$$ $$224$$ $$-0.22485$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1290.2.a.i

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} - q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + 4q^{13} - 4q^{14} + q^{15} + q^{16} - q^{18} + O(q^{20})$$ ## 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. 