# Properties

 Label 1290g Number of curves $4$ Conductor $1290$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 1290g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1290.h4 1290g1 $$[1, 0, 1, -108, 118]$$ $$137467988281/72562500$$ $$72562500$$ $$$$ $$480$$ $$0.19989$$ $$\Gamma_0(N)$$-optimal
1290.h3 1290g2 $$[1, 0, 1, -1358, 19118]$$ $$276670733768281/336980250$$ $$336980250$$ $$$$ $$960$$ $$0.54647$$
1290.h2 1290g3 $$[1, 0, 1, -4983, -135782]$$ $$13679527032530281/381633600$$ $$381633600$$ $$$$ $$1440$$ $$0.74920$$
1290.h1 1290g4 $$[1, 0, 1, -5183, -124342]$$ $$15393836938735081/2275690697640$$ $$2275690697640$$ $$$$ $$2880$$ $$1.0958$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1290.2.a.g

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 