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Properties

Label 17.1.371...761.1
Degree $17$
Signature $[1, 8]$
Discriminant $3.710\times 10^{25}$
Root discriminant \(31.92\)
Ramified prime $1571$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{17}$ (as 17T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 5*x^15 + 9*x^14 - 13*x^13 + 17*x^12 + 56*x^11 - 62*x^10 + 55*x^9 - 535*x^8 + 181*x^7 - 1027*x^6 + 661*x^5 - 1041*x^4 + 166*x^3 - 416*x^2 - 88*x - 16)
 
gp: K = bnfinit(y^17 - y^16 + 5*y^15 + 9*y^14 - 13*y^13 + 17*y^12 + 56*y^11 - 62*y^10 + 55*y^9 - 535*y^8 + 181*y^7 - 1027*y^6 + 661*y^5 - 1041*y^4 + 166*y^3 - 416*y^2 - 88*y - 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^16 + 5*x^15 + 9*x^14 - 13*x^13 + 17*x^12 + 56*x^11 - 62*x^10 + 55*x^9 - 535*x^8 + 181*x^7 - 1027*x^6 + 661*x^5 - 1041*x^4 + 166*x^3 - 416*x^2 - 88*x - 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^16 + 5*x^15 + 9*x^14 - 13*x^13 + 17*x^12 + 56*x^11 - 62*x^10 + 55*x^9 - 535*x^8 + 181*x^7 - 1027*x^6 + 661*x^5 - 1041*x^4 + 166*x^3 - 416*x^2 - 88*x - 16)
 

\( x^{17} - x^{16} + 5 x^{15} + 9 x^{14} - 13 x^{13} + 17 x^{12} + 56 x^{11} - 62 x^{10} + 55 x^{9} - 535 x^{8} + 181 x^{7} - 1027 x^{6} + 661 x^{5} - 1041 x^{4} + 166 x^{3} - 416 x^{2} + \cdots - 16 \) 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:  $[1, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(37103038784156796454937761\) \(\medspace = 1571^{8}\) 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:  \(31.92\)
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))
 
Ramified primes:   \(1571\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{11}-\frac{1}{16}a^{10}+\frac{3}{32}a^{9}-\frac{1}{16}a^{7}+\frac{1}{16}a^{6}-\frac{5}{32}a^{5}+\frac{1}{8}a^{4}-\frac{13}{32}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{16}a^{7}-\frac{1}{32}a^{6}-\frac{3}{16}a^{5}-\frac{5}{32}a^{4}-\frac{7}{16}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{64}a^{11}-\frac{1}{64}a^{10}+\frac{1}{64}a^{7}-\frac{5}{64}a^{6}+\frac{1}{64}a^{5}-\frac{9}{64}a^{4}-\frac{5}{32}a^{3}-\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{128}a^{14}-\frac{1}{64}a^{12}-\frac{1}{64}a^{11}-\frac{1}{128}a^{10}+\frac{1}{128}a^{8}-\frac{1}{32}a^{7}+\frac{7}{32}a^{6}+\frac{3}{16}a^{5}-\frac{19}{128}a^{4}+\frac{11}{64}a^{3}-\frac{7}{16}a^{2}+\frac{3}{16}a-\frac{1}{8}$, $\frac{1}{88064}a^{15}+\frac{135}{88064}a^{14}-\frac{59}{44032}a^{13}+\frac{333}{22016}a^{12}+\frac{917}{88064}a^{11}+\frac{3677}{88064}a^{10}+\frac{2673}{88064}a^{9}+\frac{9859}{88064}a^{8}+\frac{563}{22016}a^{7}-\frac{513}{5504}a^{6}+\frac{20753}{88064}a^{5}-\frac{11467}{88064}a^{4}+\frac{5677}{44032}a^{3}-\frac{2673}{5504}a^{2}-\frac{1817}{11008}a+\frac{1413}{5504}$, $\frac{1}{90806841344}a^{16}+\frac{161025}{45403420672}a^{15}+\frac{63666343}{90806841344}a^{14}-\frac{230362431}{45403420672}a^{13}-\frac{870664175}{90806841344}a^{12}-\frac{102756003}{22701710336}a^{11}+\frac{79105103}{2837713792}a^{10}+\frac{755216999}{45403420672}a^{9}-\frac{8022653443}{90806841344}a^{8}-\frac{4841956483}{22701710336}a^{7}+\frac{4677974369}{90806841344}a^{6}-\frac{16201311}{65993344}a^{5}+\frac{15286423313}{90806841344}a^{4}-\frac{15050594729}{45403420672}a^{3}+\frac{5596936081}{11350855168}a^{2}-\frac{617877433}{11350855168}a-\frac{1140399705}{5675427584}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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:  $8$
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:   $\frac{1263890223}{90806841344}a^{16}-\frac{368431735}{22701710336}a^{15}+\frac{5552907155}{90806841344}a^{14}+\frac{5060038285}{45403420672}a^{13}-\frac{22502983849}{90806841344}a^{12}+\frac{5293638533}{45403420672}a^{11}+\frac{33529381575}{45403420672}a^{10}-\frac{11609718905}{11350855168}a^{9}+\frac{23474970181}{90806841344}a^{8}-\frac{174912431355}{22701710336}a^{7}+\frac{342683486831}{90806841344}a^{6}-\frac{402379634909}{45403420672}a^{5}+\frac{1483090968221}{90806841344}a^{4}-\frac{186583266361}{45403420672}a^{3}+\frac{1861725561}{263973376}a^{2}-\frac{25030950349}{11350855168}a-\frac{2711154041}{5675427584}$, $\frac{2909689321}{90806841344}a^{16}-\frac{792071107}{22701710336}a^{15}+\frac{14549391117}{90806841344}a^{14}+\frac{12527119059}{45403420672}a^{13}-\frac{41219145951}{90806841344}a^{12}+\frac{24973908367}{45403420672}a^{11}+\frac{80739467149}{45403420672}a^{10}-\frac{1532108369}{709428448}a^{9}+\frac{158369635275}{90806841344}a^{8}-\frac{389953041941}{22701710336}a^{7}+\frac{670168407977}{90806841344}a^{6}-\frac{1454765525423}{45403420672}a^{5}+\frac{2186920230435}{90806841344}a^{4}-\frac{1495763962487}{45403420672}a^{3}+\frac{84364052757}{11350855168}a^{2}-\frac{142541336403}{11350855168}a-\frac{12400498039}{5675427584}$, $\frac{1255924841}{90806841344}a^{16}-\frac{716381609}{45403420672}a^{15}+\frac{6502932739}{90806841344}a^{14}+\frac{5261771157}{45403420672}a^{13}-\frac{17902278551}{90806841344}a^{12}+\frac{764672159}{2837713792}a^{11}+\frac{16979866051}{22701710336}a^{10}-\frac{45438685555}{45403420672}a^{9}+\frac{82282129881}{90806841344}a^{8}-\frac{167201595671}{22701710336}a^{7}+\frac{320893482697}{90806841344}a^{6}-\frac{7801890035}{527946752}a^{5}+\frac{962898037189}{90806841344}a^{4}-\frac{723529286693}{45403420672}a^{3}+\frac{37096068609}{11350855168}a^{2}-\frac{68839759517}{11350855168}a-\frac{5373308469}{5675427584}$, $\frac{1515981015}{90806841344}a^{16}-\frac{41904517}{2837713792}a^{15}+\frac{6992995999}{90806841344}a^{14}+\frac{7369788881}{45403420672}a^{13}-\frac{19819821841}{90806841344}a^{12}+\frac{9279601603}{45403420672}a^{11}+\frac{46260402789}{45403420672}a^{10}-\frac{21547776667}{22701710336}a^{9}+\frac{43971974417}{90806841344}a^{8}-\frac{194183868975}{22701710336}a^{7}+\frac{195403880663}{90806841344}a^{6}-\frac{15579460117}{1055893504}a^{5}+\frac{827331380769}{90806841344}a^{4}-\frac{599290488869}{45403420672}a^{3}-\frac{2044670813}{11350855168}a^{2}-\frac{56259623617}{11350855168}a-\frac{4454962629}{5675427584}$, $\frac{66658853}{22701710336}a^{16}-\frac{3667459}{354714224}a^{15}+\frac{496928861}{22701710336}a^{14}-\frac{46210189}{11350855168}a^{13}-\frac{2514831251}{22701710336}a^{12}+\frac{1938177881}{11350855168}a^{11}+\frac{1214071583}{11350855168}a^{10}-\frac{3909252377}{5675427584}a^{9}+\frac{14798416115}{22701710336}a^{8}-\frac{8731724141}{5675427584}a^{7}+\frac{93484012581}{22701710336}a^{6}-\frac{48860143173}{11350855168}a^{5}+\frac{146145271395}{22701710336}a^{4}-\frac{82828743439}{11350855168}a^{3}+\frac{10295820425}{2837713792}a^{2}+\frac{1972142653}{2837713792}a+\frac{248243153}{1418856896}$, $\frac{538113551}{90806841344}a^{16}-\frac{556237079}{45403420672}a^{15}+\frac{3677497429}{90806841344}a^{14}+\frac{844230963}{45403420672}a^{13}-\frac{10656879217}{90806841344}a^{12}+\frac{1330280321}{5675427584}a^{11}+\frac{3627300233}{22701710336}a^{10}-\frac{32535395325}{45403420672}a^{9}+\frac{96013317519}{90806841344}a^{8}-\frac{84767997313}{22701710336}a^{7}+\frac{389351733807}{90806841344}a^{6}-\frac{212331615179}{22701710336}a^{5}+\frac{957190484899}{90806841344}a^{4}-\frac{567814734947}{45403420672}a^{3}+\frac{105225605751}{11350855168}a^{2}-\frac{50104960459}{11350855168}a-\frac{2709711827}{5675427584}$, $\frac{167692833}{22701710336}a^{16}+\frac{9288515}{1418856896}a^{15}+\frac{524814425}{22701710336}a^{14}+\frac{1397050695}{11350855168}a^{13}+\frac{584873353}{22701710336}a^{12}-\frac{1264244019}{11350855168}a^{11}+\frac{5214826099}{11350855168}a^{10}+\frac{1698142279}{5675427584}a^{9}-\frac{11609254281}{22701710336}a^{8}-\frac{22756913753}{5675427584}a^{7}-\frac{143138205375}{22701710336}a^{6}-\frac{59494587241}{11350855168}a^{5}-\frac{56535605657}{22701710336}a^{4}+\frac{93100856189}{11350855168}a^{3}+\frac{6160161509}{2837713792}a^{2}+\frac{14487954425}{2837713792}a+\frac{199675357}{1418856896}$, $\frac{278980369}{90806841344}a^{16}-\frac{18152607}{11350855168}a^{15}+\frac{1005097729}{90806841344}a^{14}+\frac{1581799791}{45403420672}a^{13}-\frac{3944167975}{90806841344}a^{12}-\frac{554525367}{45403420672}a^{11}+\frac{7750850311}{45403420672}a^{10}-\frac{5366175123}{22701710336}a^{9}-\frac{24111045345}{90806841344}a^{8}-\frac{43105856305}{22701710336}a^{7}-\frac{98736247087}{90806841344}a^{6}-\frac{130015131213}{45403420672}a^{5}-\frac{81147186913}{90806841344}a^{4}-\frac{64297775851}{45403420672}a^{3}-\frac{34548106763}{11350855168}a^{2}-\frac{2215740703}{11350855168}a-\frac{9917600395}{5675427584}$ 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:  \( 8867681.2165 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 8867681.2165 \cdot 1}{2\cdot\sqrt{37103038784156796454937761}}\cr\approx \mathstrut & 3.5362607363 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 5*x^15 + 9*x^14 - 13*x^13 + 17*x^12 + 56*x^11 - 62*x^10 + 55*x^9 - 535*x^8 + 181*x^7 - 1027*x^6 + 661*x^5 - 1041*x^4 + 166*x^3 - 416*x^2 - 88*x - 16)
 
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 - x^16 + 5*x^15 + 9*x^14 - 13*x^13 + 17*x^12 + 56*x^11 - 62*x^10 + 55*x^9 - 535*x^8 + 181*x^7 - 1027*x^6 + 661*x^5 - 1041*x^4 + 166*x^3 - 416*x^2 - 88*x - 16, 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 - x^16 + 5*x^15 + 9*x^14 - 13*x^13 + 17*x^12 + 56*x^11 - 62*x^10 + 55*x^9 - 535*x^8 + 181*x^7 - 1027*x^6 + 661*x^5 - 1041*x^4 + 166*x^3 - 416*x^2 - 88*x - 16);
 
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 - x^16 + 5*x^15 + 9*x^14 - 13*x^13 + 17*x^12 + 56*x^11 - 62*x^10 + 55*x^9 - 535*x^8 + 181*x^7 - 1027*x^6 + 661*x^5 - 1041*x^4 + 166*x^3 - 416*x^2 - 88*x - 16);
 
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

$D_{17}$ (as 17T2):

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 solvable group of order 34
The 10 conjugacy class representatives for $D_{17}$
Character table for $D_{17}$

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]
 

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{8}{,}\,{\href{/padicField/2.1.0.1}{1} }$ $17$ $17$ $17$ ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }$ $17$ $17$ $17$ $17$ ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }$ $17$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $17$

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
\(1571\) Copy content Toggle raw display $\Q_{1571}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$