# Properties

 Label 690d Number of curves $2$ Conductor $690$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 690d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.d2 690d1 $$[1, 1, 0, -12, -36]$$ $$-217081801/285660$$ $$-285660$$ $$$$ $$144$$ $$-0.26022$$ $$\Gamma_0(N)$$-optimal
690.d1 690d2 $$[1, 1, 0, -242, -1554]$$ $$1577505447721/838350$$ $$838350$$ $$$$ $$288$$ $$0.086349$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form690.2.a.d

sage: E.q_eigenform(10)

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