# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 690c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.c2 690c1 $$[1, 1, 0, -22777, -90852059]$$ $$-1306902141891515161/3564268498800000000$$ $$-3564268498800000000$$ $$$$ $$17280$$ $$2.2387$$ $$\Gamma_0(N)$$-optimal
690.c1 690c2 $$[1, 1, 0, -3172057, -2148591611]$$ $$3529773792266261468365081/50841342773437500000$$ $$50841342773437500000$$ $$$$ $$34560$$ $$2.5853$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form690.2.a.c

sage: E.q_eigenform(10)

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