Normalized defining polynomial
\( x^{17} - 7 x^{16} + 30 x^{15} - 61 x^{14} + 115 x^{13} - 158 x^{12} + 94 x^{11} - 111 x^{10} + 268 x^{9} - 411 x^{8} + 465 x^{7} - 473 x^{6} + 314 x^{5} - 109 x^{4} + 446 x^{3} - 498 x^{2} + \cdots + 1 \)
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: | \(102143502423134353014546241\) \(\medspace = 1783^{8}\) | 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: | \(1783\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{4}{9}a^{5}-\frac{2}{9}a^{3}-\frac{4}{9}a^{2}+\frac{2}{9}a-\frac{1}{9}$, $\frac{1}{585}a^{14}+\frac{1}{117}a^{13}-\frac{7}{585}a^{12}-\frac{16}{585}a^{11}-\frac{11}{195}a^{10}+\frac{1}{39}a^{9}+\frac{86}{585}a^{8}-\frac{14}{65}a^{7}+\frac{209}{585}a^{6}+\frac{43}{585}a^{5}-\frac{131}{585}a^{4}-\frac{31}{117}a^{3}-\frac{89}{195}a^{2}-\frac{8}{65}a-\frac{74}{585}$, $\frac{1}{33345}a^{15}-\frac{11}{33345}a^{14}+\frac{1018}{33345}a^{13}+\frac{83}{1235}a^{12}-\frac{1724}{11115}a^{11}-\frac{4202}{33345}a^{10}-\frac{241}{3705}a^{9}+\frac{5258}{33345}a^{8}-\frac{296}{6669}a^{7}+\frac{599}{33345}a^{6}+\frac{137}{2565}a^{5}+\frac{58}{585}a^{4}-\frac{1733}{3705}a^{3}+\frac{1945}{6669}a^{2}+\frac{647}{3705}a-\frac{12011}{33345}$, $\frac{1}{11256194389635}a^{16}-\frac{355937}{3752064796545}a^{15}-\frac{54261082}{288620368965}a^{14}-\frac{594246340459}{11256194389635}a^{13}+\frac{36095198188}{750412959309}a^{12}+\frac{129922992988}{865861106895}a^{11}+\frac{308476329944}{11256194389635}a^{10}+\frac{972736546493}{11256194389635}a^{9}-\frac{158864838419}{1250688265515}a^{8}+\frac{149421815261}{750412959309}a^{7}-\frac{24117750937}{1250688265515}a^{6}+\frac{67069126507}{592431283665}a^{5}-\frac{1854497751497}{3752064796545}a^{4}+\frac{396662797030}{2251238877927}a^{3}-\frac{228569659978}{2251238877927}a^{2}-\frac{3077124090674}{11256194389635}a+\frac{10701895817}{45571637205}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{5863905898}{1250688265515}a^{16}-\frac{9704367676}{288620368965}a^{15}+\frac{557322844952}{3752064796545}a^{14}-\frac{91594850986}{288620368965}a^{13}+\frac{773922298562}{1250688265515}a^{12}-\frac{345261745844}{416896088505}a^{11}+\frac{2751674618114}{3752064796545}a^{10}-\frac{462321824834}{1250688265515}a^{9}+\frac{444391318648}{288620368965}a^{8}-\frac{1622184073573}{750412959309}a^{7}+\frac{6643413994522}{3752064796545}a^{6}-\frac{14980054117802}{3752064796545}a^{5}+\frac{126103945576}{138965362835}a^{4}-\frac{423069432334}{1250688265515}a^{3}+\frac{2036804280002}{750412959309}a^{2}-\frac{3255445821331}{1250688265515}a+\frac{470833347871}{750412959309}$, $\frac{71378165509}{11256194389635}a^{16}-\frac{57337845478}{1250688265515}a^{15}+\frac{146049848963}{750412959309}a^{14}-\frac{897681749447}{2251238877927}a^{13}+\frac{2651108806898}{3752064796545}a^{12}-\frac{931363875869}{865861106895}a^{11}+\frac{1442030338270}{2251238877927}a^{10}-\frac{7401137865196}{11256194389635}a^{9}+\frac{7916801648557}{3752064796545}a^{8}-\frac{8664413302153}{3752064796545}a^{7}+\frac{6291529576234}{3752064796545}a^{6}-\frac{37925873038319}{11256194389635}a^{5}+\frac{14975030920123}{3752064796545}a^{4}+\frac{7524960924866}{11256194389635}a^{3}+\frac{8832489143047}{11256194389635}a^{2}-\frac{50820261224813}{11256194389635}a+\frac{7694366723333}{11256194389635}$, $\frac{81801458693}{11256194389635}a^{16}-\frac{147820091264}{3752064796545}a^{15}+\frac{2104032371}{13165139637}a^{14}-\frac{2448155683829}{11256194389635}a^{13}+\frac{2191038575047}{3752064796545}a^{12}-\frac{704233813961}{2251238877927}a^{11}+\frac{272196606647}{2251238877927}a^{10}-\frac{1007152803049}{2251238877927}a^{9}+\frac{296860119488}{750412959309}a^{8}-\frac{447160785823}{250137653103}a^{7}+\frac{1624010512628}{750412959309}a^{6}-\frac{2316543928544}{2251238877927}a^{5}-\frac{9014849320234}{3752064796545}a^{4}-\frac{14316658645007}{11256194389635}a^{3}+\frac{6297001022836}{2251238877927}a^{2}+\frac{3115941875186}{11256194389635}a-\frac{10889106621554}{11256194389635}$, $\frac{82063861624}{11256194389635}a^{16}-\frac{205442026037}{3752064796545}a^{15}+\frac{922913574602}{3752064796545}a^{14}-\frac{6333470320627}{11256194389635}a^{13}+\frac{4067350788347}{3752064796545}a^{12}-\frac{17615003086826}{11256194389635}a^{11}+\frac{13059160554677}{11256194389635}a^{10}-\frac{1666069906712}{2251238877927}a^{9}+\frac{556597928488}{416896088505}a^{8}-\frac{683504258414}{288620368965}a^{7}+\frac{279442645937}{96206789655}a^{6}-\frac{28367684610647}{11256194389635}a^{5}+\frac{2417125808191}{3752064796545}a^{4}+\frac{23210006330909}{11256194389635}a^{3}-\frac{8910156594782}{11256194389635}a^{2}-\frac{651722045981}{592431283665}a+\frac{7157153495564}{11256194389635}$, $\frac{49492709762}{11256194389635}a^{16}-\frac{139199033053}{3752064796545}a^{15}+\frac{34555916269}{197477094555}a^{14}-\frac{390200239628}{865861106895}a^{13}+\frac{3241141069648}{3752064796545}a^{12}-\frac{3114867073670}{2251238877927}a^{11}+\frac{15169445910286}{11256194389635}a^{10}-\frac{11299969586681}{11256194389635}a^{9}+\frac{2378680973606}{1250688265515}a^{8}-\frac{12952462646524}{3752064796545}a^{7}+\frac{1764077892482}{416896088505}a^{6}-\frac{10248264291074}{2251238877927}a^{5}+\frac{15221907670493}{3752064796545}a^{4}-\frac{21919582690001}{11256194389635}a^{3}+\frac{26809457099111}{11256194389635}a^{2}-\frac{54671565486958}{11256194389635}a+\frac{31387161237277}{11256194389635}$, $\frac{22506869119}{750412959309}a^{16}-\frac{230516089046}{1250688265515}a^{15}+\frac{922294011359}{1250688265515}a^{14}-\frac{4371616048094}{3752064796545}a^{13}+\frac{2883000255353}{1250688265515}a^{12}-\frac{9181532378923}{3752064796545}a^{11}+\frac{17325653803}{288620368965}a^{10}-\frac{8004489601403}{3752064796545}a^{9}+\frac{415574920204}{83379217701}a^{8}-\frac{2531653120181}{416896088505}a^{7}+\frac{7071369709868}{1250688265515}a^{6}-\frac{22284657150016}{3752064796545}a^{5}+\frac{1473901525411}{1250688265515}a^{4}+\frac{2541458876071}{3752064796545}a^{3}+\frac{2276348233118}{197477094555}a^{2}-\frac{12355848015142}{3752064796545}a-\frac{3985860476693}{3752064796545}$, $\frac{40612104491}{3752064796545}a^{16}-\frac{85826518787}{1250688265515}a^{15}+\frac{345113190428}{1250688265515}a^{14}-\frac{90420074396}{197477094555}a^{13}+\frac{1083348191162}{1250688265515}a^{12}-\frac{3922698358438}{3752064796545}a^{11}+\frac{618686660548}{3752064796545}a^{10}-\frac{3802953393272}{3752064796545}a^{9}+\frac{3236703254452}{1250688265515}a^{8}-\frac{1153713246568}{416896088505}a^{7}+\frac{723594129004}{250137653103}a^{6}-\frac{10895790478531}{3752064796545}a^{5}+\frac{1708982369104}{1250688265515}a^{4}+\frac{43026324038}{750412959309}a^{3}+\frac{19403006951651}{3752064796545}a^{2}-\frac{10680890292688}{3752064796545}a+\frac{273932692651}{3752064796545}$, $a$ | 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
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 4818947.77807 \cdot 1}{2\cdot\sqrt{102143502423134353014546241}}\cr\approx \mathstrut & 1.15820595653 \end{aligned}\]
Galois group
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 $\Q$. |
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.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(1783\) | $\Q_{1783}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |