# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 1290j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1290.k2 1290j1 $$[1, 1, 1, -86, 239]$$ $$70393838689/8062500$$ $$8062500$$ $$$$ $$288$$ $$0.057441$$ $$\Gamma_0(N)$$-optimal
1290.k1 1290j2 $$[1, 1, 1, -1336, 18239]$$ $$263732349218689/4160250$$ $$4160250$$ $$$$ $$576$$ $$0.40401$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1290.2.a.j

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} + q^{8} + q^{9} - q^{10} + 2q^{11} - q^{12} - 2q^{13} + 2q^{14} + q^{15} + q^{16} + 4q^{17} + q^{18} - 2q^{19} + O(q^{20})$$ ## 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. 