Normalized defining polynomial
\( x^{18} - 7 x^{17} + 17 x^{16} + 17 x^{15} - 935 x^{14} + 799 x^{13} + 9231 x^{12} - 41463 x^{11} + 192780 x^{10} + 291686 x^{9} - 390014 x^{8} + 6132223 x^{7} + \cdots + 113422599 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(40254497110927943179349807054456171205137\) \(\medspace = 17^{33}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(180.23\) | 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: | $17^{543/272}\approx 286.0053344539552$ | ||
Ramified primes: | \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}-\frac{1}{9}a^{5}-\frac{2}{9}a^{4}-\frac{4}{9}a^{3}-\frac{2}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{2}{9}a^{4}+\frac{2}{9}a^{3}-\frac{1}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{27}a^{13}-\frac{1}{27}a^{11}-\frac{1}{27}a^{10}+\frac{1}{27}a^{9}+\frac{1}{27}a^{8}-\frac{1}{27}a^{7}-\frac{1}{27}a^{6}+\frac{4}{27}a^{5}-\frac{1}{3}a^{4}-\frac{1}{27}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{27}a^{14}-\frac{1}{27}a^{12}-\frac{1}{27}a^{11}+\frac{1}{27}a^{10}+\frac{1}{27}a^{9}-\frac{1}{27}a^{8}-\frac{1}{27}a^{7}+\frac{4}{27}a^{6}-\frac{10}{27}a^{4}+\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{27}a^{15}-\frac{1}{27}a^{12}+\frac{1}{9}a^{7}-\frac{1}{27}a^{6}+\frac{1}{9}a^{5}-\frac{2}{9}a^{4}-\frac{1}{27}a^{3}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{16}-\frac{1}{27}a^{11}-\frac{1}{27}a^{10}+\frac{1}{27}a^{9}+\frac{4}{27}a^{8}-\frac{2}{27}a^{7}+\frac{2}{27}a^{6}-\frac{2}{27}a^{5}-\frac{10}{27}a^{4}-\frac{7}{27}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{70\!\cdots\!37}a^{17}+\frac{33\!\cdots\!81}{70\!\cdots\!37}a^{16}+\frac{11\!\cdots\!11}{70\!\cdots\!37}a^{15}+\frac{63\!\cdots\!48}{70\!\cdots\!37}a^{14}-\frac{74\!\cdots\!10}{70\!\cdots\!37}a^{13}+\frac{26\!\cdots\!00}{70\!\cdots\!37}a^{12}+\frac{10\!\cdots\!69}{87\!\cdots\!77}a^{11}-\frac{12\!\cdots\!35}{78\!\cdots\!93}a^{10}+\frac{53\!\cdots\!04}{78\!\cdots\!93}a^{9}+\frac{48\!\cdots\!93}{70\!\cdots\!37}a^{8}+\frac{57\!\cdots\!46}{70\!\cdots\!37}a^{7}+\frac{18\!\cdots\!23}{70\!\cdots\!37}a^{6}+\frac{60\!\cdots\!55}{70\!\cdots\!37}a^{5}-\frac{34\!\cdots\!51}{70\!\cdots\!37}a^{4}-\frac{50\!\cdots\!74}{70\!\cdots\!37}a^{3}-\frac{39\!\cdots\!75}{23\!\cdots\!79}a^{2}+\frac{53\!\cdots\!13}{26\!\cdots\!31}a+\frac{18\!\cdots\!57}{26\!\cdots\!31}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $9$ | 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: | $\frac{28\!\cdots\!65}{70\!\cdots\!37}a^{17}-\frac{30\!\cdots\!07}{70\!\cdots\!37}a^{16}+\frac{16\!\cdots\!38}{70\!\cdots\!37}a^{15}+\frac{21\!\cdots\!13}{70\!\cdots\!37}a^{14}-\frac{26\!\cdots\!58}{70\!\cdots\!37}a^{13}+\frac{17\!\cdots\!96}{70\!\cdots\!37}a^{12}+\frac{20\!\cdots\!86}{78\!\cdots\!93}a^{11}-\frac{44\!\cdots\!28}{87\!\cdots\!77}a^{10}+\frac{80\!\cdots\!56}{78\!\cdots\!93}a^{9}-\frac{25\!\cdots\!88}{70\!\cdots\!37}a^{8}-\frac{13\!\cdots\!23}{70\!\cdots\!37}a^{7}+\frac{12\!\cdots\!13}{70\!\cdots\!37}a^{6}-\frac{10\!\cdots\!12}{70\!\cdots\!37}a^{5}-\frac{56\!\cdots\!68}{70\!\cdots\!37}a^{4}+\frac{24\!\cdots\!19}{70\!\cdots\!37}a^{3}-\frac{31\!\cdots\!89}{23\!\cdots\!79}a^{2}+\frac{43\!\cdots\!28}{26\!\cdots\!31}a-\frac{40\!\cdots\!57}{26\!\cdots\!31}$, $\frac{14\!\cdots\!08}{70\!\cdots\!37}a^{17}-\frac{94\!\cdots\!00}{70\!\cdots\!37}a^{16}-\frac{26\!\cdots\!52}{70\!\cdots\!37}a^{15}+\frac{61\!\cdots\!63}{70\!\cdots\!37}a^{14}-\frac{51\!\cdots\!15}{70\!\cdots\!37}a^{13}+\frac{18\!\cdots\!11}{70\!\cdots\!37}a^{12}-\frac{23\!\cdots\!48}{87\!\cdots\!77}a^{11}-\frac{26\!\cdots\!90}{26\!\cdots\!31}a^{10}+\frac{64\!\cdots\!56}{78\!\cdots\!93}a^{9}-\frac{16\!\cdots\!70}{70\!\cdots\!37}a^{8}+\frac{25\!\cdots\!70}{70\!\cdots\!37}a^{7}+\frac{18\!\cdots\!36}{70\!\cdots\!37}a^{6}-\frac{16\!\cdots\!83}{70\!\cdots\!37}a^{5}+\frac{48\!\cdots\!61}{70\!\cdots\!37}a^{4}-\frac{80\!\cdots\!68}{70\!\cdots\!37}a^{3}+\frac{32\!\cdots\!68}{23\!\cdots\!79}a^{2}-\frac{26\!\cdots\!36}{26\!\cdots\!31}a+\frac{13\!\cdots\!15}{26\!\cdots\!31}$, $\frac{52\!\cdots\!62}{70\!\cdots\!37}a^{17}-\frac{23\!\cdots\!05}{70\!\cdots\!37}a^{16}+\frac{87\!\cdots\!01}{70\!\cdots\!37}a^{15}+\frac{14\!\cdots\!78}{70\!\cdots\!37}a^{14}-\frac{14\!\cdots\!89}{70\!\cdots\!37}a^{13}+\frac{14\!\cdots\!58}{70\!\cdots\!37}a^{12}+\frac{48\!\cdots\!71}{78\!\cdots\!93}a^{11}-\frac{29\!\cdots\!58}{78\!\cdots\!93}a^{10}-\frac{23\!\cdots\!53}{78\!\cdots\!93}a^{9}-\frac{15\!\cdots\!02}{70\!\cdots\!37}a^{8}-\frac{84\!\cdots\!17}{70\!\cdots\!37}a^{7}+\frac{61\!\cdots\!47}{70\!\cdots\!37}a^{6}-\frac{21\!\cdots\!33}{70\!\cdots\!37}a^{5}-\frac{39\!\cdots\!30}{70\!\cdots\!37}a^{4}+\frac{10\!\cdots\!26}{70\!\cdots\!37}a^{3}-\frac{93\!\cdots\!80}{23\!\cdots\!79}a^{2}+\frac{82\!\cdots\!66}{26\!\cdots\!31}a-\frac{78\!\cdots\!76}{26\!\cdots\!31}$, $\frac{14\!\cdots\!68}{23\!\cdots\!79}a^{17}-\frac{48\!\cdots\!94}{23\!\cdots\!79}a^{16}-\frac{76\!\cdots\!26}{23\!\cdots\!79}a^{15}+\frac{64\!\cdots\!81}{23\!\cdots\!79}a^{14}-\frac{10\!\cdots\!42}{23\!\cdots\!79}a^{13}-\frac{37\!\cdots\!98}{23\!\cdots\!79}a^{12}+\frac{39\!\cdots\!64}{78\!\cdots\!93}a^{11}+\frac{26\!\cdots\!80}{78\!\cdots\!93}a^{10}+\frac{39\!\cdots\!83}{78\!\cdots\!93}a^{9}+\frac{99\!\cdots\!25}{23\!\cdots\!79}a^{8}+\frac{23\!\cdots\!35}{23\!\cdots\!79}a^{7}+\frac{77\!\cdots\!79}{23\!\cdots\!79}a^{6}+\frac{13\!\cdots\!18}{23\!\cdots\!79}a^{5}+\frac{16\!\cdots\!44}{23\!\cdots\!79}a^{4}+\frac{37\!\cdots\!67}{23\!\cdots\!79}a^{3}-\frac{24\!\cdots\!29}{78\!\cdots\!93}a^{2}+\frac{56\!\cdots\!03}{29\!\cdots\!59}a-\frac{38\!\cdots\!22}{87\!\cdots\!77}$, $\frac{16\!\cdots\!84}{70\!\cdots\!37}a^{17}-\frac{10\!\cdots\!01}{70\!\cdots\!37}a^{16}+\frac{19\!\cdots\!39}{70\!\cdots\!37}a^{15}+\frac{49\!\cdots\!15}{70\!\cdots\!37}a^{14}-\frac{15\!\cdots\!48}{70\!\cdots\!37}a^{13}+\frac{43\!\cdots\!47}{70\!\cdots\!37}a^{12}+\frac{21\!\cdots\!02}{87\!\cdots\!77}a^{11}-\frac{61\!\cdots\!60}{78\!\cdots\!93}a^{10}+\frac{29\!\cdots\!97}{78\!\cdots\!93}a^{9}+\frac{65\!\cdots\!20}{70\!\cdots\!37}a^{8}-\frac{60\!\cdots\!94}{70\!\cdots\!37}a^{7}+\frac{81\!\cdots\!88}{70\!\cdots\!37}a^{6}-\frac{41\!\cdots\!46}{70\!\cdots\!37}a^{5}-\frac{12\!\cdots\!74}{70\!\cdots\!37}a^{4}+\frac{44\!\cdots\!99}{70\!\cdots\!37}a^{3}-\frac{51\!\cdots\!20}{23\!\cdots\!79}a^{2}+\frac{42\!\cdots\!67}{26\!\cdots\!31}a-\frac{51\!\cdots\!32}{26\!\cdots\!31}$, $\frac{13\!\cdots\!79}{23\!\cdots\!79}a^{17}-\frac{75\!\cdots\!09}{23\!\cdots\!79}a^{16}+\frac{63\!\cdots\!24}{23\!\cdots\!79}a^{15}+\frac{71\!\cdots\!23}{23\!\cdots\!79}a^{14}-\frac{12\!\cdots\!61}{23\!\cdots\!79}a^{13}-\frac{86\!\cdots\!10}{23\!\cdots\!79}a^{12}+\frac{21\!\cdots\!91}{29\!\cdots\!59}a^{11}-\frac{11\!\cdots\!30}{78\!\cdots\!93}a^{10}+\frac{15\!\cdots\!41}{26\!\cdots\!31}a^{9}+\frac{79\!\cdots\!60}{23\!\cdots\!79}a^{8}-\frac{25\!\cdots\!56}{23\!\cdots\!79}a^{7}+\frac{49\!\cdots\!24}{23\!\cdots\!79}a^{6}+\frac{51\!\cdots\!99}{23\!\cdots\!79}a^{5}-\frac{14\!\cdots\!82}{23\!\cdots\!79}a^{4}+\frac{38\!\cdots\!94}{23\!\cdots\!79}a^{3}-\frac{12\!\cdots\!56}{78\!\cdots\!93}a^{2}+\frac{23\!\cdots\!30}{29\!\cdots\!59}a+\frac{14\!\cdots\!06}{87\!\cdots\!77}$, $\frac{17\!\cdots\!87}{23\!\cdots\!79}a^{17}-\frac{18\!\cdots\!11}{23\!\cdots\!79}a^{16}+\frac{97\!\cdots\!02}{23\!\cdots\!79}a^{15}-\frac{31\!\cdots\!91}{23\!\cdots\!79}a^{14}-\frac{58\!\cdots\!71}{23\!\cdots\!79}a^{13}+\frac{40\!\cdots\!72}{23\!\cdots\!79}a^{12}+\frac{11\!\cdots\!58}{29\!\cdots\!59}a^{11}-\frac{83\!\cdots\!36}{26\!\cdots\!31}a^{10}+\frac{69\!\cdots\!17}{26\!\cdots\!31}a^{9}-\frac{17\!\cdots\!07}{23\!\cdots\!79}a^{8}+\frac{52\!\cdots\!75}{23\!\cdots\!79}a^{7}-\frac{57\!\cdots\!22}{23\!\cdots\!79}a^{6}+\frac{37\!\cdots\!66}{23\!\cdots\!79}a^{5}+\frac{18\!\cdots\!31}{23\!\cdots\!79}a^{4}-\frac{43\!\cdots\!10}{23\!\cdots\!79}a^{3}+\frac{26\!\cdots\!71}{78\!\cdots\!93}a^{2}-\frac{22\!\cdots\!59}{87\!\cdots\!77}a+\frac{16\!\cdots\!65}{87\!\cdots\!77}$, $\frac{17\!\cdots\!62}{70\!\cdots\!37}a^{17}-\frac{19\!\cdots\!36}{70\!\cdots\!37}a^{16}+\frac{71\!\cdots\!14}{70\!\cdots\!37}a^{15}+\frac{40\!\cdots\!65}{70\!\cdots\!37}a^{14}-\frac{22\!\cdots\!44}{70\!\cdots\!37}a^{13}+\frac{82\!\cdots\!25}{70\!\cdots\!37}a^{12}+\frac{28\!\cdots\!47}{78\!\cdots\!93}a^{11}-\frac{22\!\cdots\!54}{78\!\cdots\!93}a^{10}+\frac{17\!\cdots\!52}{26\!\cdots\!31}a^{9}+\frac{47\!\cdots\!46}{70\!\cdots\!37}a^{8}-\frac{63\!\cdots\!90}{70\!\cdots\!37}a^{7}+\frac{10\!\cdots\!84}{70\!\cdots\!37}a^{6}-\frac{84\!\cdots\!30}{70\!\cdots\!37}a^{5}-\frac{42\!\cdots\!57}{70\!\cdots\!37}a^{4}+\frac{12\!\cdots\!12}{70\!\cdots\!37}a^{3}-\frac{70\!\cdots\!28}{23\!\cdots\!79}a^{2}+\frac{65\!\cdots\!89}{26\!\cdots\!31}a-\frac{43\!\cdots\!26}{26\!\cdots\!31}$, $\frac{39\!\cdots\!01}{70\!\cdots\!37}a^{17}-\frac{14\!\cdots\!09}{70\!\cdots\!37}a^{16}+\frac{28\!\cdots\!88}{70\!\cdots\!37}a^{15}+\frac{22\!\cdots\!83}{70\!\cdots\!37}a^{14}-\frac{31\!\cdots\!64}{70\!\cdots\!37}a^{13}-\frac{67\!\cdots\!28}{70\!\cdots\!37}a^{12}+\frac{15\!\cdots\!35}{78\!\cdots\!93}a^{11}-\frac{22\!\cdots\!56}{78\!\cdots\!93}a^{10}-\frac{17\!\cdots\!47}{78\!\cdots\!93}a^{9}+\frac{45\!\cdots\!73}{70\!\cdots\!37}a^{8}-\frac{41\!\cdots\!41}{70\!\cdots\!37}a^{7}+\frac{31\!\cdots\!34}{70\!\cdots\!37}a^{6}+\frac{93\!\cdots\!50}{70\!\cdots\!37}a^{5}-\frac{30\!\cdots\!90}{70\!\cdots\!37}a^{4}+\frac{60\!\cdots\!77}{70\!\cdots\!37}a^{3}-\frac{33\!\cdots\!17}{23\!\cdots\!79}a^{2}+\frac{27\!\cdots\!78}{26\!\cdots\!31}a-\frac{19\!\cdots\!36}{26\!\cdots\!31}$ (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: | \( 45413380405400 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 45413380405400 \cdot 1}{2\cdot\sqrt{40254497110927943179349807054456171205137}}\cr\approx \mathstrut & 1.09962744387092 \end{aligned}\] (assuming GRH)
Galois group
$\PGL(2,17)$ (as 18T468):
A non-solvable group of order 4896 |
The 19 conjugacy class representatives for $\PGL(2,17)$ |
Character table for $\PGL(2,17)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 36 sibling: | 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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{9}$ | $16{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $18$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | $18$ | $16{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
In the table, R denotes a ramified prime. 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 |
---|---|---|---|---|---|---|---|
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
17.17.33.11 | $x^{17} + 867 x^{2} + 2890 x + 17$ | $17$ | $1$ | $33$ | $F_{17}$ | $[33/16]_{16}$ |
Additional information
This field is associated with the 17-division points on any elliptic curve in the isogeny class 17.a1.