# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 690f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.f3 690f1 $$[1, 0, 1, -13, 8]$$ $$217081801/88320$$ $$88320$$ $$$$ $$64$$ $$-0.35322$$ $$\Gamma_0(N)$$-optimal
690.f2 690f2 $$[1, 0, 1, -93, -344]$$ $$87587538121/1904400$$ $$1904400$$ $$[2, 2]$$ $$128$$ $$-0.0066500$$
690.f1 690f3 $$[1, 0, 1, -1473, -21872]$$ $$353108405631241/172500$$ $$172500$$ $$$$ $$256$$ $$0.33992$$
690.f4 690f4 $$[1, 0, 1, 7, -1024]$$ $$46268279/453342420$$ $$-453342420$$ $$$$ $$256$$ $$0.33992$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form690.2.a.f

sage: E.q_eigenform(10)

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