# Properties

 Label 690g Number of curves $4$ Conductor $690$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 690g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.h4 690g1 $$[1, 1, 1, -4491, -207687]$$ $$-10017490085065009/12502381363200$$ $$-12502381363200$$ $$$$ $$1792$$ $$1.2065$$ $$\Gamma_0(N)$$-optimal
690.h3 690g2 $$[1, 1, 1, -86411, -9808711]$$ $$71356102305927901489/35540674560000$$ $$35540674560000$$ $$[2, 2]$$ $$3584$$ $$1.5531$$
690.h1 690g3 $$[1, 1, 1, -1382411, -626186311]$$ $$292169767125103365085489/72534787200$$ $$72534787200$$ $$$$ $$7168$$ $$1.8996$$
690.h2 690g4 $$[1, 1, 1, -101131, -6258247]$$ $$114387056741228939569/49503729150000000$$ $$49503729150000000$$ $$$$ $$7168$$ $$1.8996$$

## Rank

sage: E.rank()

The elliptic curves in class 690g have rank $$0$$.

## Complex multiplication

The elliptic curves in class 690g do not have complex multiplication.

## Modular form690.2.a.g

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 