Properties

Label 17.1.102...241.1
Degree $17$
Signature $[1, 8]$
Discriminant $1.021\times 10^{26}$
Root discriminant \(33.88\)
Ramified prime $1783$
Class number $1$
Class group trivial
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 - 7*x^16 + 30*x^15 - 61*x^14 + 115*x^13 - 158*x^12 + 94*x^11 - 111*x^10 + 268*x^9 - 411*x^8 + 465*x^7 - 473*x^6 + 314*x^5 - 109*x^4 + 446*x^3 - 498*x^2 + 175*x + 1)
 
gp: K = bnfinit(y^17 - 7*y^16 + 30*y^15 - 61*y^14 + 115*y^13 - 158*y^12 + 94*y^11 - 111*y^10 + 268*y^9 - 411*y^8 + 465*y^7 - 473*y^6 + 314*y^5 - 109*y^4 + 446*y^3 - 498*y^2 + 175*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 7*x^16 + 30*x^15 - 61*x^14 + 115*x^13 - 158*x^12 + 94*x^11 - 111*x^10 + 268*x^9 - 411*x^8 + 465*x^7 - 473*x^6 + 314*x^5 - 109*x^4 + 446*x^3 - 498*x^2 + 175*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 7*x^16 + 30*x^15 - 61*x^14 + 115*x^13 - 158*x^12 + 94*x^11 - 111*x^10 + 268*x^9 - 411*x^8 + 465*x^7 - 473*x^6 + 314*x^5 - 109*x^4 + 446*x^3 - 498*x^2 + 175*x + 1)
 

\( x^{17} - 7 x^{16} + 30 x^{15} - 61 x^{14} + 115 x^{13} - 158 x^{12} + 94 x^{11} - 111 x^{10} + 268 x^{9} - 411 x^{8} + 465 x^{7} - 473 x^{6} + 314 x^{5} - 109 x^{4} + 446 x^{3} - 498 x^{2} + \cdots + 1 \) 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:  $[1, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(102143502423134353014546241\) \(\medspace = 1783^{8}\) 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:  \(33.88\)
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:  $1783^{1/2}\approx 42.22558466143482$
Ramified primes:   \(1783\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{4}{9}a^{5}-\frac{2}{9}a^{3}-\frac{4}{9}a^{2}+\frac{2}{9}a-\frac{1}{9}$, $\frac{1}{585}a^{14}+\frac{1}{117}a^{13}-\frac{7}{585}a^{12}-\frac{16}{585}a^{11}-\frac{11}{195}a^{10}+\frac{1}{39}a^{9}+\frac{86}{585}a^{8}-\frac{14}{65}a^{7}+\frac{209}{585}a^{6}+\frac{43}{585}a^{5}-\frac{131}{585}a^{4}-\frac{31}{117}a^{3}-\frac{89}{195}a^{2}-\frac{8}{65}a-\frac{74}{585}$, $\frac{1}{33345}a^{15}-\frac{11}{33345}a^{14}+\frac{1018}{33345}a^{13}+\frac{83}{1235}a^{12}-\frac{1724}{11115}a^{11}-\frac{4202}{33345}a^{10}-\frac{241}{3705}a^{9}+\frac{5258}{33345}a^{8}-\frac{296}{6669}a^{7}+\frac{599}{33345}a^{6}+\frac{137}{2565}a^{5}+\frac{58}{585}a^{4}-\frac{1733}{3705}a^{3}+\frac{1945}{6669}a^{2}+\frac{647}{3705}a-\frac{12011}{33345}$, $\frac{1}{11256194389635}a^{16}-\frac{355937}{3752064796545}a^{15}-\frac{54261082}{288620368965}a^{14}-\frac{594246340459}{11256194389635}a^{13}+\frac{36095198188}{750412959309}a^{12}+\frac{129922992988}{865861106895}a^{11}+\frac{308476329944}{11256194389635}a^{10}+\frac{972736546493}{11256194389635}a^{9}-\frac{158864838419}{1250688265515}a^{8}+\frac{149421815261}{750412959309}a^{7}-\frac{24117750937}{1250688265515}a^{6}+\frac{67069126507}{592431283665}a^{5}-\frac{1854497751497}{3752064796545}a^{4}+\frac{396662797030}{2251238877927}a^{3}-\frac{228569659978}{2251238877927}a^{2}-\frac{3077124090674}{11256194389635}a+\frac{10701895817}{45571637205}$ 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:  $3$

Class group and class number

Trivial group, which has order $1$

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:   $\frac{5863905898}{1250688265515}a^{16}-\frac{9704367676}{288620368965}a^{15}+\frac{557322844952}{3752064796545}a^{14}-\frac{91594850986}{288620368965}a^{13}+\frac{773922298562}{1250688265515}a^{12}-\frac{345261745844}{416896088505}a^{11}+\frac{2751674618114}{3752064796545}a^{10}-\frac{462321824834}{1250688265515}a^{9}+\frac{444391318648}{288620368965}a^{8}-\frac{1622184073573}{750412959309}a^{7}+\frac{6643413994522}{3752064796545}a^{6}-\frac{14980054117802}{3752064796545}a^{5}+\frac{126103945576}{138965362835}a^{4}-\frac{423069432334}{1250688265515}a^{3}+\frac{2036804280002}{750412959309}a^{2}-\frac{3255445821331}{1250688265515}a+\frac{470833347871}{750412959309}$, $\frac{71378165509}{11256194389635}a^{16}-\frac{57337845478}{1250688265515}a^{15}+\frac{146049848963}{750412959309}a^{14}-\frac{897681749447}{2251238877927}a^{13}+\frac{2651108806898}{3752064796545}a^{12}-\frac{931363875869}{865861106895}a^{11}+\frac{1442030338270}{2251238877927}a^{10}-\frac{7401137865196}{11256194389635}a^{9}+\frac{7916801648557}{3752064796545}a^{8}-\frac{8664413302153}{3752064796545}a^{7}+\frac{6291529576234}{3752064796545}a^{6}-\frac{37925873038319}{11256194389635}a^{5}+\frac{14975030920123}{3752064796545}a^{4}+\frac{7524960924866}{11256194389635}a^{3}+\frac{8832489143047}{11256194389635}a^{2}-\frac{50820261224813}{11256194389635}a+\frac{7694366723333}{11256194389635}$, $\frac{81801458693}{11256194389635}a^{16}-\frac{147820091264}{3752064796545}a^{15}+\frac{2104032371}{13165139637}a^{14}-\frac{2448155683829}{11256194389635}a^{13}+\frac{2191038575047}{3752064796545}a^{12}-\frac{704233813961}{2251238877927}a^{11}+\frac{272196606647}{2251238877927}a^{10}-\frac{1007152803049}{2251238877927}a^{9}+\frac{296860119488}{750412959309}a^{8}-\frac{447160785823}{250137653103}a^{7}+\frac{1624010512628}{750412959309}a^{6}-\frac{2316543928544}{2251238877927}a^{5}-\frac{9014849320234}{3752064796545}a^{4}-\frac{14316658645007}{11256194389635}a^{3}+\frac{6297001022836}{2251238877927}a^{2}+\frac{3115941875186}{11256194389635}a-\frac{10889106621554}{11256194389635}$, $\frac{82063861624}{11256194389635}a^{16}-\frac{205442026037}{3752064796545}a^{15}+\frac{922913574602}{3752064796545}a^{14}-\frac{6333470320627}{11256194389635}a^{13}+\frac{4067350788347}{3752064796545}a^{12}-\frac{17615003086826}{11256194389635}a^{11}+\frac{13059160554677}{11256194389635}a^{10}-\frac{1666069906712}{2251238877927}a^{9}+\frac{556597928488}{416896088505}a^{8}-\frac{683504258414}{288620368965}a^{7}+\frac{279442645937}{96206789655}a^{6}-\frac{28367684610647}{11256194389635}a^{5}+\frac{2417125808191}{3752064796545}a^{4}+\frac{23210006330909}{11256194389635}a^{3}-\frac{8910156594782}{11256194389635}a^{2}-\frac{651722045981}{592431283665}a+\frac{7157153495564}{11256194389635}$, $\frac{49492709762}{11256194389635}a^{16}-\frac{139199033053}{3752064796545}a^{15}+\frac{34555916269}{197477094555}a^{14}-\frac{390200239628}{865861106895}a^{13}+\frac{3241141069648}{3752064796545}a^{12}-\frac{3114867073670}{2251238877927}a^{11}+\frac{15169445910286}{11256194389635}a^{10}-\frac{11299969586681}{11256194389635}a^{9}+\frac{2378680973606}{1250688265515}a^{8}-\frac{12952462646524}{3752064796545}a^{7}+\frac{1764077892482}{416896088505}a^{6}-\frac{10248264291074}{2251238877927}a^{5}+\frac{15221907670493}{3752064796545}a^{4}-\frac{21919582690001}{11256194389635}a^{3}+\frac{26809457099111}{11256194389635}a^{2}-\frac{54671565486958}{11256194389635}a+\frac{31387161237277}{11256194389635}$, $\frac{22506869119}{750412959309}a^{16}-\frac{230516089046}{1250688265515}a^{15}+\frac{922294011359}{1250688265515}a^{14}-\frac{4371616048094}{3752064796545}a^{13}+\frac{2883000255353}{1250688265515}a^{12}-\frac{9181532378923}{3752064796545}a^{11}+\frac{17325653803}{288620368965}a^{10}-\frac{8004489601403}{3752064796545}a^{9}+\frac{415574920204}{83379217701}a^{8}-\frac{2531653120181}{416896088505}a^{7}+\frac{7071369709868}{1250688265515}a^{6}-\frac{22284657150016}{3752064796545}a^{5}+\frac{1473901525411}{1250688265515}a^{4}+\frac{2541458876071}{3752064796545}a^{3}+\frac{2276348233118}{197477094555}a^{2}-\frac{12355848015142}{3752064796545}a-\frac{3985860476693}{3752064796545}$, $\frac{40612104491}{3752064796545}a^{16}-\frac{85826518787}{1250688265515}a^{15}+\frac{345113190428}{1250688265515}a^{14}-\frac{90420074396}{197477094555}a^{13}+\frac{1083348191162}{1250688265515}a^{12}-\frac{3922698358438}{3752064796545}a^{11}+\frac{618686660548}{3752064796545}a^{10}-\frac{3802953393272}{3752064796545}a^{9}+\frac{3236703254452}{1250688265515}a^{8}-\frac{1153713246568}{416896088505}a^{7}+\frac{723594129004}{250137653103}a^{6}-\frac{10895790478531}{3752064796545}a^{5}+\frac{1708982369104}{1250688265515}a^{4}+\frac{43026324038}{750412959309}a^{3}+\frac{19403006951651}{3752064796545}a^{2}-\frac{10680890292688}{3752064796545}a+\frac{273932692651}{3752064796545}$, $a$ Copy content Toggle raw display
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:  \( 4818947.77807 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 4818947.77807 \cdot 1}{2\cdot\sqrt{102143502423134353014546241}}\cr\approx \mathstrut & 1.15820595653 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^17 - 7*x^16 + 30*x^15 - 61*x^14 + 115*x^13 - 158*x^12 + 94*x^11 - 111*x^10 + 268*x^9 - 411*x^8 + 465*x^7 - 473*x^6 + 314*x^5 - 109*x^4 + 446*x^3 - 498*x^2 + 175*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 - 7*x^16 + 30*x^15 - 61*x^14 + 115*x^13 - 158*x^12 + 94*x^11 - 111*x^10 + 268*x^9 - 411*x^8 + 465*x^7 - 473*x^6 + 314*x^5 - 109*x^4 + 446*x^3 - 498*x^2 + 175*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 - 7*x^16 + 30*x^15 - 61*x^14 + 115*x^13 - 158*x^12 + 94*x^11 - 111*x^10 + 268*x^9 - 411*x^8 + 465*x^7 - 473*x^6 + 314*x^5 - 109*x^4 + 446*x^3 - 498*x^2 + 175*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 - 7*x^16 + 30*x^15 - 61*x^14 + 115*x^13 - 158*x^12 + 94*x^11 - 111*x^10 + 268*x^9 - 411*x^8 + 465*x^7 - 473*x^6 + 314*x^5 - 109*x^4 + 446*x^3 - 498*x^2 + 175*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

$D_{17}$ (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
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $17$ ${\href{/padicField/3.2.0.1}{2} }^{8}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.2.0.1}{2} }^{8}{,}\,{\href{/padicField/5.1.0.1}{1} }$ $17$ ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }$ $17$ ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $17$ $17$ $17$ $17$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $17$

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
\(1783\) Copy content Toggle raw display $\Q_{1783}$$x$$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}$