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Properties

Label 17.1.160...625.1
Degree $17$
Signature $[1, 8]$
Discriminant $1.607\times 10^{25}$
Root discriminant \(30.39\)
Ramified primes $5,283$
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 - 4*x^16 + x^15 + 16*x^14 - 30*x^13 - 2*x^12 + 88*x^11 + 143*x^10 + 133*x^9 - 515*x^8 - 803*x^7 - 68*x^6 + 1252*x^5 + 2972*x^4 + 2855*x^3 + 1505*x^2 + 550*x + 125)
 
gp: K = bnfinit(y^17 - 4*y^16 + y^15 + 16*y^14 - 30*y^13 - 2*y^12 + 88*y^11 + 143*y^10 + 133*y^9 - 515*y^8 - 803*y^7 - 68*y^6 + 1252*y^5 + 2972*y^4 + 2855*y^3 + 1505*y^2 + 550*y + 125, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 4*x^16 + x^15 + 16*x^14 - 30*x^13 - 2*x^12 + 88*x^11 + 143*x^10 + 133*x^9 - 515*x^8 - 803*x^7 - 68*x^6 + 1252*x^5 + 2972*x^4 + 2855*x^3 + 1505*x^2 + 550*x + 125);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 4*x^16 + x^15 + 16*x^14 - 30*x^13 - 2*x^12 + 88*x^11 + 143*x^10 + 133*x^9 - 515*x^8 - 803*x^7 - 68*x^6 + 1252*x^5 + 2972*x^4 + 2855*x^3 + 1505*x^2 + 550*x + 125)
 

\( x^{17} - 4 x^{16} + x^{15} + 16 x^{14} - 30 x^{13} - 2 x^{12} + 88 x^{11} + 143 x^{10} + 133 x^{9} - 515 x^{8} - 803 x^{7} - 68 x^{6} + 1252 x^{5} + 2972 x^{4} + 2855 x^{3} + 1505 x^{2} + \cdots + 125 \) 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:   \(16071318903145633687890625\) \(\medspace = 5^{8}\cdot 283^{8}\) 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:  \(30.39\)
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:   \(5\), \(283\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{2}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}$, $\frac{1}{5}a^{9}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{6}+\frac{1}{5}a^{2}$, $\frac{1}{25}a^{11}-\frac{2}{25}a^{10}+\frac{2}{25}a^{9}+\frac{2}{25}a^{8}+\frac{2}{25}a^{7}+\frac{4}{25}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{9}{25}a^{3}+\frac{2}{5}a$, $\frac{1}{25}a^{12}-\frac{2}{25}a^{10}+\frac{1}{25}a^{9}+\frac{1}{25}a^{8}-\frac{2}{25}a^{7}+\frac{3}{25}a^{6}+\frac{1}{5}a^{5}-\frac{11}{25}a^{4}-\frac{2}{25}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{125}a^{13}-\frac{2}{125}a^{11}+\frac{11}{125}a^{10}+\frac{6}{125}a^{9}-\frac{2}{125}a^{8}-\frac{2}{125}a^{7}-\frac{1}{5}a^{6}-\frac{11}{125}a^{5}+\frac{3}{125}a^{4}+\frac{8}{25}a^{3}-\frac{1}{25}a^{2}+\frac{1}{5}a$, $\frac{1}{125}a^{14}-\frac{2}{125}a^{12}+\frac{1}{125}a^{11}+\frac{1}{125}a^{10}+\frac{3}{125}a^{9}+\frac{3}{125}a^{8}+\frac{1}{25}a^{7}-\frac{1}{125}a^{6}+\frac{53}{125}a^{5}+\frac{3}{25}a^{4}+\frac{1}{25}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{875}a^{15}+\frac{2}{875}a^{14}+\frac{3}{875}a^{13}+\frac{1}{125}a^{12}+\frac{3}{875}a^{11}-\frac{1}{175}a^{10}-\frac{8}{125}a^{9}+\frac{81}{875}a^{8}+\frac{24}{875}a^{7}-\frac{229}{875}a^{6}-\frac{184}{875}a^{5}+\frac{8}{175}a^{4}+\frac{8}{25}a^{3}-\frac{1}{35}a^{2}-\frac{1}{35}a-\frac{2}{7}$, $\frac{1}{2615944069375}a^{16}+\frac{218937184}{2615944069375}a^{15}+\frac{368743109}{373706295625}a^{14}+\frac{737489092}{523188813875}a^{13}+\frac{7919246736}{523188813875}a^{12}-\frac{35805577492}{2615944069375}a^{11}+\frac{196162750347}{2615944069375}a^{10}-\frac{26640288041}{2615944069375}a^{9}-\frac{46749711864}{523188813875}a^{8}+\frac{47954499897}{523188813875}a^{7}+\frac{710493631017}{2615944069375}a^{6}-\frac{229449530337}{2615944069375}a^{5}+\frac{171732759551}{2615944069375}a^{4}+\frac{244902547024}{523188813875}a^{3}+\frac{33135306798}{523188813875}a^{2}-\frac{52235576909}{104637762775}a-\frac{4281699832}{20927552555}$ 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:  $5$

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:   $\frac{637787931}{74741259125}a^{16}-\frac{19735843466}{523188813875}a^{15}+\frac{11717674662}{523188813875}a^{14}+\frac{70392364229}{523188813875}a^{13}-\frac{23053849088}{74741259125}a^{12}+\frac{32519538196}{523188813875}a^{11}+\frac{411351224579}{523188813875}a^{10}+\frac{14773405428}{14948251825}a^{9}+\frac{195075083156}{523188813875}a^{8}-\frac{2577709140988}{523188813875}a^{7}-\frac{2331344948939}{523188813875}a^{6}+\frac{1230023036703}{523188813875}a^{5}+\frac{45790807938}{4185510511}a^{4}+\frac{302154989063}{14948251825}a^{3}+\frac{248973906373}{20927552555}a^{2}+\frac{74183967756}{20927552555}a+\frac{3563748152}{4185510511}$, $\frac{13166742333}{2615944069375}a^{16}-\frac{53402672318}{2615944069375}a^{15}+\frac{2871733867}{373706295625}a^{14}+\frac{37450696694}{523188813875}a^{13}-\frac{74800927391}{523188813875}a^{12}+\frac{37313929019}{2615944069375}a^{11}+\frac{993139224656}{2615944069375}a^{10}+\frac{1892906104857}{2615944069375}a^{9}+\frac{405726948453}{523188813875}a^{8}-\frac{258072843687}{104637762775}a^{7}-\frac{11520362899249}{2615944069375}a^{6}-\frac{2748375903186}{2615944069375}a^{5}+\frac{17611993464788}{2615944069375}a^{4}+\frac{8695071255982}{523188813875}a^{3}+\frac{8193945281974}{523188813875}a^{2}+\frac{614710345213}{104637762775}a+\frac{12004814404}{20927552555}$, $\frac{2547137324}{373706295625}a^{16}-\frac{67002087273}{2615944069375}a^{15}-\frac{28336310671}{2615944069375}a^{14}+\frac{88188732467}{523188813875}a^{13}-\frac{19696775886}{74741259125}a^{12}-\frac{311794279516}{2615944069375}a^{11}+\frac{2531811900446}{2615944069375}a^{10}+\frac{258786396636}{373706295625}a^{9}+\frac{418340941386}{523188813875}a^{8}-\frac{2432027134891}{523188813875}a^{7}-\frac{15943929525444}{2615944069375}a^{6}+\frac{9019097649209}{2615944069375}a^{5}+\frac{27103434402048}{2615944069375}a^{4}+\frac{1518813599661}{74741259125}a^{3}+\frac{6515865863629}{523188813875}a^{2}-\frac{149264678022}{104637762775}a-\frac{38200000086}{20927552555}$, $\frac{1665828917}{523188813875}a^{16}-\frac{1619984766}{104637762775}a^{15}+\frac{957469116}{74741259125}a^{14}+\frac{30931238523}{523188813875}a^{13}-\frac{18102605051}{104637762775}a^{12}+\frac{12183650057}{104637762775}a^{11}+\frac{164619280093}{523188813875}a^{10}+\frac{23782956304}{523188813875}a^{9}+\frac{123107642674}{523188813875}a^{8}-\frac{1176423255136}{523188813875}a^{7}-\frac{339794484966}{523188813875}a^{6}+\frac{1128233959542}{523188813875}a^{5}+\frac{1575932467653}{523188813875}a^{4}+\frac{649956666822}{104637762775}a^{3}-\frac{28896962441}{104637762775}a^{2}-\frac{24991638104}{20927552555}a-\frac{2972644096}{4185510511}$, $\frac{3564399001}{2615944069375}a^{16}-\frac{12800355101}{2615944069375}a^{15}-\frac{4652689257}{2615944069375}a^{14}+\frac{13536537366}{523188813875}a^{13}-\frac{15913871858}{523188813875}a^{12}-\frac{17238456946}{373706295625}a^{11}+\frac{440018412297}{2615944069375}a^{10}+\frac{653868099669}{2615944069375}a^{9}+\frac{49279812934}{523188813875}a^{8}-\frac{47658341973}{74741259125}a^{7}-\frac{3955173682518}{2615944069375}a^{6}+\frac{343300369028}{2615944069375}a^{5}+\frac{6118823155576}{2615944069375}a^{4}+\frac{1965682977514}{523188813875}a^{3}+\frac{2155287959173}{523188813875}a^{2}+\frac{19845053193}{14948251825}a+\frac{4062812458}{20927552555}$, $\frac{1407393157}{373706295625}a^{16}-\frac{9024916747}{373706295625}a^{15}+\frac{19306766671}{373706295625}a^{14}-\frac{139502178}{14948251825}a^{13}-\frac{12714319477}{74741259125}a^{12}+\frac{124650242656}{373706295625}a^{11}-\frac{29838771061}{373706295625}a^{10}+\frac{99067161333}{373706295625}a^{9}-\frac{33161347594}{74741259125}a^{8}-\frac{149390155078}{74741259125}a^{7}+\frac{699414079859}{373706295625}a^{6}+\frac{547092284296}{373706295625}a^{5}+\frac{1088517791337}{373706295625}a^{4}+\frac{72258209548}{74741259125}a^{3}-\frac{354827466699}{74741259125}a^{2}-\frac{13096115938}{14948251825}a-\frac{3710357944}{2989650365}$, $\frac{26682634334}{523188813875}a^{16}-\frac{115269991652}{523188813875}a^{15}+\frac{8698387733}{74741259125}a^{14}+\frac{420847695591}{523188813875}a^{13}-\frac{950062980436}{523188813875}a^{12}+\frac{215767701533}{523188813875}a^{11}+\frac{2408079371463}{523188813875}a^{10}+\frac{2918593963764}{523188813875}a^{9}+\frac{2461194816832}{523188813875}a^{8}-\frac{14669613961787}{523188813875}a^{7}-\frac{3393405721614}{104637762775}a^{6}+\frac{5281809739604}{523188813875}a^{5}+\frac{32271323889529}{523188813875}a^{4}+\frac{13627198799863}{104637762775}a^{3}+\frac{10342688082707}{104637762775}a^{2}+\frac{730359633127}{20927552555}a+\frac{61559023803}{4185510511}$, $\frac{28692107627}{2615944069375}a^{16}-\frac{165623700257}{2615944069375}a^{15}+\frac{306890546141}{2615944069375}a^{14}-\frac{144266471}{523188813875}a^{13}-\frac{198988531614}{523188813875}a^{12}+\frac{234493920708}{373706295625}a^{11}+\frac{180546534134}{2615944069375}a^{10}+\frac{3066531320213}{2615944069375}a^{9}-\frac{419668759404}{523188813875}a^{8}-\frac{358934565213}{74741259125}a^{7}+\frac{58372524419}{2615944069375}a^{6}+\frac{4845469231446}{2615944069375}a^{5}+\frac{28928181591077}{2615944069375}a^{4}+\frac{6137692964538}{523188813875}a^{3}+\frac{2308513454971}{523188813875}a^{2}+\frac{21767117471}{14948251825}a+\frac{14326819786}{20927552555}$ 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:  \( 2534195.05633 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 2534195.05633 \cdot 1}{2\cdot\sqrt{16071318903145633687890625}}\cr\approx \mathstrut & 1.53551204241 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^16 + x^15 + 16*x^14 - 30*x^13 - 2*x^12 + 88*x^11 + 143*x^10 + 133*x^9 - 515*x^8 - 803*x^7 - 68*x^6 + 1252*x^5 + 2972*x^4 + 2855*x^3 + 1505*x^2 + 550*x + 125)
 
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 - 4*x^16 + x^15 + 16*x^14 - 30*x^13 - 2*x^12 + 88*x^11 + 143*x^10 + 133*x^9 - 515*x^8 - 803*x^7 - 68*x^6 + 1252*x^5 + 2972*x^4 + 2855*x^3 + 1505*x^2 + 550*x + 125, 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 - 4*x^16 + x^15 + 16*x^14 - 30*x^13 - 2*x^12 + 88*x^11 + 143*x^10 + 133*x^9 - 515*x^8 - 803*x^7 - 68*x^6 + 1252*x^5 + 2972*x^4 + 2855*x^3 + 1505*x^2 + 550*x + 125);
 
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 - 4*x^16 + x^15 + 16*x^14 - 30*x^13 - 2*x^12 + 88*x^11 + 143*x^10 + 133*x^9 - 515*x^8 - 803*x^7 - 68*x^6 + 1252*x^5 + 2972*x^4 + 2855*x^3 + 1505*x^2 + 550*x + 125);
 
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 $17$ $17$ R ${\href{/padicField/7.2.0.1}{2} }^{8}{,}\,{\href{/padicField/7.1.0.1}{1} }$ $17$ ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }$ $17$ ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $17$ ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $17$ $17$ $17$ $17$ $17$ $17$

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
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
\(283\) Copy content Toggle raw display $\Q_{283}$$x$$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}$