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Properties

Label 17.5.478...988.1
Degree $17$
Signature $[5, 6]$
Discriminant $4.781\times 10^{25}$
Root discriminant \(32.40\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{17}$ (as 17T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 - 2*x^15 + 9*x^14 - 8*x^13 - 6*x^12 + 22*x^11 - 20*x^10 - 4*x^9 + 23*x^8 - 20*x^7 + 7*x^6 + x^5 - 11*x^4 + 11*x^3 - x^2 - 3*x + 1)
 
gp: K = bnfinit(y^17 - 2*y^16 - 2*y^15 + 9*y^14 - 8*y^13 - 6*y^12 + 22*y^11 - 20*y^10 - 4*y^9 + 23*y^8 - 20*y^7 + 7*y^6 + y^5 - 11*y^4 + 11*y^3 - y^2 - 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 - 2*x^15 + 9*x^14 - 8*x^13 - 6*x^12 + 22*x^11 - 20*x^10 - 4*x^9 + 23*x^8 - 20*x^7 + 7*x^6 + x^5 - 11*x^4 + 11*x^3 - x^2 - 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 - 2*x^15 + 9*x^14 - 8*x^13 - 6*x^12 + 22*x^11 - 20*x^10 - 4*x^9 + 23*x^8 - 20*x^7 + 7*x^6 + x^5 - 11*x^4 + 11*x^3 - x^2 - 3*x + 1)
 

\( x^{17} - 2 x^{16} - 2 x^{15} + 9 x^{14} - 8 x^{13} - 6 x^{12} + 22 x^{11} - 20 x^{10} - 4 x^{9} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[5, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(47814524918055414168841988\) \(\medspace = 2^{2}\cdot 71\cdot 113\cdot 398023\cdot 3743302227733793\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(32.40\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{2/3}71^{1/2}113^{1/2}398023^{1/2}3743302227733793^{1/2}\approx 5488284177909.384$
Ramified primes:   \(2\), \(71\), \(113\), \(398023\), \(3743302227733793\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{11953\!\cdots\!10497}$)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17}a^{16}+\frac{3}{17}a^{15}-\frac{4}{17}a^{14}+\frac{6}{17}a^{13}+\frac{5}{17}a^{12}+\frac{2}{17}a^{11}-\frac{2}{17}a^{10}+\frac{4}{17}a^{9}-\frac{1}{17}a^{8}+\frac{1}{17}a^{7}+\frac{2}{17}a^{6}+\frac{1}{17}a^{4}-\frac{6}{17}a^{3}-\frac{2}{17}a^{2}+\frac{6}{17}a-\frac{7}{17}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $10$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a$, $\frac{1}{17}a^{16}-\frac{14}{17}a^{15}+\frac{13}{17}a^{14}+\frac{40}{17}a^{13}-\frac{80}{17}a^{12}+\frac{36}{17}a^{11}+\frac{83}{17}a^{10}-\frac{183}{17}a^{9}+\frac{118}{17}a^{8}+\frac{86}{17}a^{7}-\frac{168}{17}a^{6}+7a^{5}-\frac{33}{17}a^{4}-\frac{23}{17}a^{3}+\frac{117}{17}a^{2}-\frac{11}{17}a-\frac{24}{17}$, $\frac{40}{17}a^{16}-\frac{67}{17}a^{15}-\frac{92}{17}a^{14}+\frac{308}{17}a^{13}-\frac{225}{17}a^{12}-\frac{243}{17}a^{11}+\frac{702}{17}a^{10}-\frac{554}{17}a^{9}-\frac{210}{17}a^{8}+\frac{652}{17}a^{7}-\frac{498}{17}a^{6}+11a^{5}-\frac{28}{17}a^{4}-\frac{342}{17}a^{3}+\frac{260}{17}a^{2}+\frac{36}{17}a-\frac{59}{17}$, $\frac{9}{17}a^{16}-\frac{7}{17}a^{15}-\frac{36}{17}a^{14}+\frac{54}{17}a^{13}+\frac{11}{17}a^{12}-\frac{101}{17}a^{11}+\frac{118}{17}a^{10}-\frac{15}{17}a^{9}-\frac{128}{17}a^{8}+\frac{128}{17}a^{7}-\frac{67}{17}a^{6}+2a^{5}+\frac{43}{17}a^{4}-\frac{207}{17}a^{3}+\frac{67}{17}a^{2}+\frac{88}{17}a-\frac{46}{17}$, $\frac{33}{17}a^{16}-\frac{71}{17}a^{15}-\frac{30}{17}a^{14}+\frac{249}{17}a^{13}-\frac{328}{17}a^{12}+\frac{32}{17}a^{11}+\frac{495}{17}a^{10}-\frac{718}{17}a^{9}+\frac{324}{17}a^{8}+\frac{237}{17}a^{7}-\frac{512}{17}a^{6}+27a^{5}-\frac{341}{17}a^{4}-\frac{45}{17}a^{3}+\frac{172}{17}a^{2}-\frac{74}{17}a+\frac{7}{17}$, $\frac{25}{17}a^{16}-\frac{44}{17}a^{15}-\frac{49}{17}a^{14}+\frac{184}{17}a^{13}-\frac{164}{17}a^{12}-\frac{86}{17}a^{11}+\frac{392}{17}a^{10}-\frac{393}{17}a^{9}+\frac{26}{17}a^{8}+\frac{263}{17}a^{7}-\frac{324}{17}a^{6}+14a^{5}-\frac{145}{17}a^{4}-\frac{116}{17}a^{3}+\frac{154}{17}a^{2}-\frac{20}{17}a-\frac{22}{17}$, $\frac{3}{17}a^{16}-\frac{8}{17}a^{15}-\frac{12}{17}a^{14}+\frac{52}{17}a^{13}-\frac{19}{17}a^{12}-\frac{113}{17}a^{11}+\frac{181}{17}a^{10}-\frac{5}{17}a^{9}-\frac{309}{17}a^{8}+\frac{360}{17}a^{7}+\frac{23}{17}a^{6}-23a^{5}+\frac{309}{17}a^{4}+\frac{16}{17}a^{3}-\frac{125}{17}a^{2}+\frac{86}{17}a-\frac{21}{17}$, $\frac{24}{17}a^{16}-\frac{30}{17}a^{15}-\frac{79}{17}a^{14}+\frac{178}{17}a^{13}-\frac{50}{17}a^{12}-\frac{258}{17}a^{11}+\frac{428}{17}a^{10}-\frac{159}{17}a^{9}-\frac{381}{17}a^{8}+\frac{483}{17}a^{7}-\frac{190}{17}a^{6}-5a^{5}+\frac{143}{17}a^{4}-\frac{280}{17}a^{3}+\frac{139}{17}a^{2}+\frac{110}{17}a-\frac{66}{17}$, $\frac{52}{17}a^{16}-\frac{184}{17}a^{15}+\frac{64}{17}a^{14}+\frac{550}{17}a^{13}-\frac{1015}{17}a^{12}+\frac{461}{17}a^{11}+\frac{1069}{17}a^{10}-\frac{2223}{17}a^{9}+\frac{1529}{17}a^{8}+\frac{375}{17}a^{7}-\frac{1511}{17}a^{6}+91a^{5}-\frac{1138}{17}a^{4}+\frac{351}{17}a^{3}+\frac{559}{17}a^{2}-\frac{334}{17}a+\frac{27}{17}$, $\frac{55}{17}a^{16}-\frac{90}{17}a^{15}-\frac{152}{17}a^{14}+\frac{449}{17}a^{13}-\frac{252}{17}a^{12}-\frac{468}{17}a^{11}+\frac{1063}{17}a^{10}-\frac{664}{17}a^{9}-\frac{565}{17}a^{8}+\frac{1109}{17}a^{7}-\frac{638}{17}a^{6}+5a^{5}+\frac{123}{17}a^{4}-\frac{568}{17}a^{3}+\frac{383}{17}a^{2}+\frac{109}{17}a-\frac{130}{17}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4697339.96204 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{6}\cdot 4697339.96204 \cdot 1}{2\cdot\sqrt{47814524918055414168841988}}\cr\approx \mathstrut & 0.668761499602 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 - 2*x^15 + 9*x^14 - 8*x^13 - 6*x^12 + 22*x^11 - 20*x^10 - 4*x^9 + 23*x^8 - 20*x^7 + 7*x^6 + x^5 - 11*x^4 + 11*x^3 - x^2 - 3*x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^17 - 2*x^16 - 2*x^15 + 9*x^14 - 8*x^13 - 6*x^12 + 22*x^11 - 20*x^10 - 4*x^9 + 23*x^8 - 20*x^7 + 7*x^6 + x^5 - 11*x^4 + 11*x^3 - x^2 - 3*x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^17 - 2*x^16 - 2*x^15 + 9*x^14 - 8*x^13 - 6*x^12 + 22*x^11 - 20*x^10 - 4*x^9 + 23*x^8 - 20*x^7 + 7*x^6 + x^5 - 11*x^4 + 11*x^3 - x^2 - 3*x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 - 2*x^15 + 9*x^14 - 8*x^13 - 6*x^12 + 22*x^11 - 20*x^10 - 4*x^9 + 23*x^8 - 20*x^7 + 7*x^6 + x^5 - 11*x^4 + 11*x^3 - x^2 - 3*x + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$S_{17}$ (as 17T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 355687428096000
The 297 conjugacy class representatives for $S_{17}$
Character table for $S_{17}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ $17$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $17$ ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ $16{,}\,{\href{/padicField/47.1.0.1}{1} }$ $16{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.0.1$x^{4} + x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.7.0.1$x^{7} + x + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
\(71\) Copy content Toggle raw display 71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.1$x^{2} + 497$$2$$1$$1$$C_2$$[\ ]_{2}$
71.6.0.1$x^{6} + x^{4} + 10 x^{3} + 13 x^{2} + 29 x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$
71.7.0.1$x^{7} + 2 x + 64$$1$$7$$0$$C_7$$[\ ]^{7}$
\(113\) Copy content Toggle raw display $\Q_{113}$$x + 110$$1$$1$$0$Trivial$[\ ]$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.3.0.1$x^{3} + 8 x + 110$$1$$3$$0$$C_3$$[\ ]^{3}$
113.3.0.1$x^{3} + 8 x + 110$$1$$3$$0$$C_3$$[\ ]^{3}$
113.4.0.1$x^{4} + 62 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
\(398023\) Copy content Toggle raw display $\Q_{398023}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{398023}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{398023}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{398023}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(3743302227733793\) Copy content Toggle raw display $\Q_{3743302227733793}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$