# Properties

 Label 17689g Number of curves $3$ Conductor $17689$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 17689g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
17689.f3 17689g1 $$[0, 1, 1, 11793, -17853]$$ $$32768/19$$ $$-105163116221611$$ $$[]$$ $$45360$$ $$1.3800$$ $$\Gamma_0(N)$$-optimal
17689.f2 17689g2 $$[0, 1, 1, -165097, -27524248]$$ $$-89915392/6859$$ $$-37963884956001571$$ $$[]$$ $$136080$$ $$1.9293$$
17689.f1 17689g3 $$[0, 1, 1, -13608737, -19327549923]$$ $$-50357871050752/19$$ $$-105163116221611$$ $$[]$$ $$408240$$ $$2.4786$$

## Rank

sage: E.rank()

The elliptic curves in class 17689g have rank $$1$$.

## Complex multiplication

The elliptic curves in class 17689g do not have complex multiplication.

## Modular form 17689.2.a.g

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.