# Properties

 Label 17.1.124...441.1 Degree $17$ Signature $[1, 8]$ Discriminant $1.249\times 10^{23}$ Root discriminant $22.84$ Ramified primes $11, 17$ Class number $1$ Class group trivial Galois group $C_{17}:C_{4}$ (as 17T3)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^16 + 6*x^15 - 5*x^14 + 7*x^13 - 17*x^12 + 24*x^11 - 16*x^10 + 42*x^9 - 83*x^8 + 82*x^7 - 72*x^6 + 37*x^5 + 24*x^4 - 10*x^3 + 14*x^2 - 2*x + 1)

gp: K = bnfinit(x^17 - 4*x^16 + 6*x^15 - 5*x^14 + 7*x^13 - 17*x^12 + 24*x^11 - 16*x^10 + 42*x^9 - 83*x^8 + 82*x^7 - 72*x^6 + 37*x^5 + 24*x^4 - 10*x^3 + 14*x^2 - 2*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 14, -10, 24, 37, -72, 82, -83, 42, -16, 24, -17, 7, -5, 6, -4, 1]);

$$x^{17} - 4 x^{16} + 6 x^{15} - 5 x^{14} + 7 x^{13} - 17 x^{12} + 24 x^{11} - 16 x^{10} + 42 x^{9} - 83 x^{8} + 82 x^{7} - 72 x^{6} + 37 x^{5} + 24 x^{4} - 10 x^{3} + 14 x^{2} - 2 x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $17$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[1, 8]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$124890250818288107857441$$$$\medspace = 11^{8}\cdot 17^{12}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $22.84$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $11, 17$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{8689217585} a^{16} - \frac{409389363}{8689217585} a^{15} + \frac{746341569}{8689217585} a^{14} + \frac{68674948}{8689217585} a^{13} + \frac{2982233818}{8689217585} a^{12} + \frac{804301212}{1737843517} a^{11} + \frac{81858777}{1737843517} a^{10} + \frac{1092336176}{8689217585} a^{9} + \frac{288687551}{8689217585} a^{8} + \frac{247093695}{1737843517} a^{7} + \frac{492792409}{1737843517} a^{6} - \frac{2557040536}{8689217585} a^{5} + \frac{833505687}{8689217585} a^{4} - \frac{67104075}{1737843517} a^{3} + \frac{2641511796}{8689217585} a^{2} - \frac{929864671}{8689217585} a + \frac{2495583072}{8689217585}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $8$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$83428.2679357$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{8}\cdot 83428.2679357 \cdot 1}{2\sqrt{124890250818288107857441}}\approx 0.573439896420$

## Galois group

$D_{17}.C_2$ (as 17T3):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 68 The 8 conjugacy class representatives for $C_{17}:C_{4}$ Character table for $C_{17}:C_{4}$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type $17$ ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }$ R $17$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $17$ $17$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

gp: idealfactors = idealprimedec(K, p); \\ get the data

gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

magma: idealfactors := Factorization(p*Integers(K)); // get the data

magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$11$$\Q_{11}$$x + 3$$1$$1$$0Trivial[\ ] 11.4.2.2x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2} 11.4.2.2x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2} 17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4} 17.4.3.2x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4} 17.4.3.2x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$