Normalized defining polynomial
\( x^{17} - 3 x^{16} + 3 x^{15} - 17 x^{14} + 27 x^{13} + 27 x^{12} + 85 x^{11} - 303 x^{10} + 141 x^{9} + \cdots - 48 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-725232993643865780590215168\) \(\medspace = -\,2^{29}\cdot 3^{38}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.02\) | 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: | not computed | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{14\!\cdots\!00}a^{16}-\frac{22\!\cdots\!21}{14\!\cdots\!00}a^{15}+\frac{35\!\cdots\!31}{14\!\cdots\!00}a^{14}-\frac{79\!\cdots\!51}{59\!\cdots\!56}a^{13}+\frac{83\!\cdots\!27}{14\!\cdots\!00}a^{12}-\frac{33\!\cdots\!59}{14\!\cdots\!00}a^{11}+\frac{54\!\cdots\!97}{14\!\cdots\!00}a^{10}+\frac{66\!\cdots\!51}{14\!\cdots\!00}a^{9}+\frac{52\!\cdots\!73}{14\!\cdots\!00}a^{8}-\frac{61\!\cdots\!49}{14\!\cdots\!00}a^{7}+\frac{44\!\cdots\!41}{14\!\cdots\!00}a^{6}+\frac{51\!\cdots\!63}{14\!\cdots\!00}a^{5}-\frac{74\!\cdots\!99}{14\!\cdots\!00}a^{4}+\frac{71\!\cdots\!99}{29\!\cdots\!80}a^{3}+\frac{15\!\cdots\!59}{59\!\cdots\!56}a^{2}+\frac{13\!\cdots\!81}{14\!\cdots\!00}a-\frac{14\!\cdots\!84}{37\!\cdots\!25}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{91\!\cdots\!07}{14\!\cdots\!00}a^{16}+\frac{26\!\cdots\!97}{14\!\cdots\!00}a^{15}-\frac{23\!\cdots\!17}{14\!\cdots\!00}a^{14}+\frac{60\!\cdots\!23}{59\!\cdots\!56}a^{13}-\frac{23\!\cdots\!89}{14\!\cdots\!00}a^{12}-\frac{29\!\cdots\!37}{14\!\cdots\!00}a^{11}-\frac{79\!\cdots\!79}{14\!\cdots\!00}a^{10}+\frac{27\!\cdots\!93}{14\!\cdots\!00}a^{9}-\frac{84\!\cdots\!11}{14\!\cdots\!00}a^{8}+\frac{10\!\cdots\!93}{14\!\cdots\!00}a^{7}-\frac{55\!\cdots\!87}{14\!\cdots\!00}a^{6}+\frac{14\!\cdots\!09}{14\!\cdots\!00}a^{5}-\frac{63\!\cdots\!07}{14\!\cdots\!00}a^{4}-\frac{14\!\cdots\!43}{29\!\cdots\!80}a^{3}-\frac{86\!\cdots\!77}{59\!\cdots\!56}a^{2}+\frac{33\!\cdots\!83}{14\!\cdots\!00}a-\frac{14\!\cdots\!12}{37\!\cdots\!25}$, $\frac{20\!\cdots\!51}{74\!\cdots\!50}a^{16}-\frac{65\!\cdots\!71}{74\!\cdots\!50}a^{15}+\frac{78\!\cdots\!31}{74\!\cdots\!50}a^{14}-\frac{13\!\cdots\!47}{29\!\cdots\!78}a^{13}+\frac{56\!\cdots\!77}{74\!\cdots\!50}a^{12}+\frac{45\!\cdots\!91}{74\!\cdots\!50}a^{11}+\frac{13\!\cdots\!97}{74\!\cdots\!50}a^{10}-\frac{59\!\cdots\!49}{74\!\cdots\!50}a^{9}+\frac{52\!\cdots\!73}{74\!\cdots\!50}a^{8}-\frac{19\!\cdots\!99}{74\!\cdots\!50}a^{7}+\frac{80\!\cdots\!41}{74\!\cdots\!50}a^{6}-\frac{26\!\cdots\!87}{74\!\cdots\!50}a^{5}+\frac{24\!\cdots\!01}{74\!\cdots\!50}a^{4}-\frac{77\!\cdots\!21}{14\!\cdots\!90}a^{3}+\frac{13\!\cdots\!63}{29\!\cdots\!78}a^{2}-\frac{61\!\cdots\!19}{74\!\cdots\!50}a+\frac{11\!\cdots\!07}{37\!\cdots\!25}$, $\frac{98\!\cdots\!29}{14\!\cdots\!00}a^{16}+\frac{32\!\cdots\!59}{14\!\cdots\!00}a^{15}-\frac{43\!\cdots\!99}{14\!\cdots\!00}a^{14}+\frac{77\!\cdots\!21}{59\!\cdots\!56}a^{13}-\frac{35\!\cdots\!83}{14\!\cdots\!00}a^{12}-\frac{90\!\cdots\!39}{14\!\cdots\!00}a^{11}-\frac{91\!\cdots\!13}{14\!\cdots\!00}a^{10}+\frac{32\!\cdots\!71}{14\!\cdots\!00}a^{9}-\frac{23\!\cdots\!17}{14\!\cdots\!00}a^{8}+\frac{18\!\cdots\!71}{14\!\cdots\!00}a^{7}-\frac{82\!\cdots\!89}{14\!\cdots\!00}a^{6}+\frac{18\!\cdots\!23}{14\!\cdots\!00}a^{5}-\frac{15\!\cdots\!29}{14\!\cdots\!00}a^{4}+\frac{40\!\cdots\!19}{29\!\cdots\!80}a^{3}-\frac{98\!\cdots\!19}{59\!\cdots\!56}a^{2}+\frac{42\!\cdots\!01}{14\!\cdots\!00}a-\frac{31\!\cdots\!64}{37\!\cdots\!25}$, $\frac{52\!\cdots\!29}{14\!\cdots\!00}a^{16}-\frac{11\!\cdots\!59}{14\!\cdots\!00}a^{15}+\frac{24\!\cdots\!99}{14\!\cdots\!00}a^{14}-\frac{30\!\cdots\!01}{59\!\cdots\!56}a^{13}+\frac{82\!\cdots\!83}{14\!\cdots\!00}a^{12}+\frac{25\!\cdots\!39}{14\!\cdots\!00}a^{11}+\frac{60\!\cdots\!13}{14\!\cdots\!00}a^{10}-\frac{14\!\cdots\!71}{14\!\cdots\!00}a^{9}-\frac{45\!\cdots\!83}{14\!\cdots\!00}a^{8}+\frac{40\!\cdots\!29}{14\!\cdots\!00}a^{7}+\frac{38\!\cdots\!89}{14\!\cdots\!00}a^{6}-\frac{72\!\cdots\!23}{14\!\cdots\!00}a^{5}+\frac{19\!\cdots\!29}{14\!\cdots\!00}a^{4}+\frac{78\!\cdots\!41}{29\!\cdots\!80}a^{3}+\frac{64\!\cdots\!87}{59\!\cdots\!56}a^{2}-\frac{19\!\cdots\!01}{14\!\cdots\!00}a+\frac{28\!\cdots\!14}{37\!\cdots\!25}$, $\frac{13\!\cdots\!77}{37\!\cdots\!25}a^{16}+\frac{38\!\cdots\!92}{37\!\cdots\!25}a^{15}-\frac{34\!\cdots\!87}{37\!\cdots\!25}a^{14}+\frac{89\!\cdots\!51}{14\!\cdots\!89}a^{13}-\frac{31\!\cdots\!29}{37\!\cdots\!25}a^{12}-\frac{40\!\cdots\!82}{37\!\cdots\!25}a^{11}-\frac{12\!\cdots\!19}{37\!\cdots\!25}a^{10}+\frac{37\!\cdots\!73}{37\!\cdots\!25}a^{9}-\frac{16\!\cdots\!71}{37\!\cdots\!25}a^{8}-\frac{52\!\cdots\!02}{37\!\cdots\!25}a^{7}-\frac{69\!\cdots\!07}{37\!\cdots\!25}a^{6}+\frac{22\!\cdots\!24}{37\!\cdots\!25}a^{5}-\frac{95\!\cdots\!77}{37\!\cdots\!25}a^{4}-\frac{23\!\cdots\!63}{74\!\cdots\!45}a^{3}-\frac{12\!\cdots\!97}{14\!\cdots\!89}a^{2}+\frac{47\!\cdots\!13}{37\!\cdots\!25}a-\frac{74\!\cdots\!03}{37\!\cdots\!25}$, $\frac{65\!\cdots\!53}{14\!\cdots\!00}a^{16}-\frac{56\!\cdots\!63}{14\!\cdots\!00}a^{15}+\frac{65\!\cdots\!43}{14\!\cdots\!00}a^{14}-\frac{43\!\cdots\!29}{59\!\cdots\!56}a^{13}+\frac{65\!\cdots\!31}{14\!\cdots\!00}a^{12}+\frac{44\!\cdots\!23}{14\!\cdots\!00}a^{11}-\frac{12\!\cdots\!59}{14\!\cdots\!00}a^{10}-\frac{55\!\cdots\!47}{14\!\cdots\!00}a^{9}+\frac{18\!\cdots\!69}{14\!\cdots\!00}a^{8}+\frac{25\!\cdots\!53}{14\!\cdots\!00}a^{7}+\frac{28\!\cdots\!73}{14\!\cdots\!00}a^{6}-\frac{17\!\cdots\!11}{14\!\cdots\!00}a^{5}+\frac{23\!\cdots\!53}{14\!\cdots\!00}a^{4}+\frac{44\!\cdots\!37}{29\!\cdots\!80}a^{3}+\frac{28\!\cdots\!51}{59\!\cdots\!56}a^{2}-\frac{70\!\cdots\!57}{14\!\cdots\!00}a+\frac{73\!\cdots\!48}{37\!\cdots\!25}$, $\frac{86\!\cdots\!91}{14\!\cdots\!00}a^{16}-\frac{22\!\cdots\!61}{14\!\cdots\!00}a^{15}+\frac{17\!\cdots\!21}{14\!\cdots\!00}a^{14}-\frac{55\!\cdots\!07}{59\!\cdots\!56}a^{13}+\frac{18\!\cdots\!57}{14\!\cdots\!00}a^{12}+\frac{30\!\cdots\!81}{14\!\cdots\!00}a^{11}+\frac{84\!\cdots\!27}{14\!\cdots\!00}a^{10}-\frac{23\!\cdots\!09}{14\!\cdots\!00}a^{9}+\frac{31\!\cdots\!43}{14\!\cdots\!00}a^{8}-\frac{12\!\cdots\!09}{14\!\cdots\!00}a^{7}+\frac{52\!\cdots\!31}{14\!\cdots\!00}a^{6}-\frac{11\!\cdots\!17}{14\!\cdots\!00}a^{5}+\frac{39\!\cdots\!91}{14\!\cdots\!00}a^{4}+\frac{12\!\cdots\!99}{29\!\cdots\!80}a^{3}+\frac{76\!\cdots\!21}{59\!\cdots\!56}a^{2}-\frac{26\!\cdots\!79}{14\!\cdots\!00}a+\frac{16\!\cdots\!06}{37\!\cdots\!25}$, $\frac{26\!\cdots\!36}{74\!\cdots\!45}a^{16}-\frac{70\!\cdots\!91}{74\!\cdots\!45}a^{15}+\frac{52\!\cdots\!66}{74\!\cdots\!45}a^{14}-\frac{86\!\cdots\!22}{14\!\cdots\!89}a^{13}+\frac{55\!\cdots\!52}{74\!\cdots\!45}a^{12}+\frac{95\!\cdots\!36}{74\!\cdots\!45}a^{11}+\frac{26\!\cdots\!97}{74\!\cdots\!45}a^{10}-\frac{71\!\cdots\!69}{74\!\cdots\!45}a^{9}+\frac{98\!\cdots\!13}{74\!\cdots\!45}a^{8}-\frac{24\!\cdots\!99}{74\!\cdots\!45}a^{7}+\frac{16\!\cdots\!91}{74\!\cdots\!45}a^{6}-\frac{37\!\cdots\!92}{74\!\cdots\!45}a^{5}+\frac{12\!\cdots\!96}{74\!\cdots\!45}a^{4}+\frac{38\!\cdots\!59}{14\!\cdots\!89}a^{3}+\frac{11\!\cdots\!27}{14\!\cdots\!89}a^{2}-\frac{83\!\cdots\!14}{74\!\cdots\!45}a+\frac{43\!\cdots\!79}{74\!\cdots\!45}$, $\frac{55\!\cdots\!56}{37\!\cdots\!25}a^{16}+\frac{69\!\cdots\!74}{37\!\cdots\!25}a^{15}-\frac{45\!\cdots\!64}{37\!\cdots\!25}a^{14}-\frac{29\!\cdots\!23}{14\!\cdots\!89}a^{13}-\frac{16\!\cdots\!88}{37\!\cdots\!25}a^{12}+\frac{54\!\cdots\!21}{37\!\cdots\!25}a^{11}+\frac{18\!\cdots\!82}{37\!\cdots\!25}a^{10}-\frac{25\!\cdots\!69}{37\!\cdots\!25}a^{9}-\frac{57\!\cdots\!62}{37\!\cdots\!25}a^{8}-\frac{35\!\cdots\!44}{37\!\cdots\!25}a^{7}+\frac{68\!\cdots\!21}{37\!\cdots\!25}a^{6}+\frac{85\!\cdots\!78}{37\!\cdots\!25}a^{5}-\frac{45\!\cdots\!44}{37\!\cdots\!25}a^{4}-\frac{31\!\cdots\!41}{74\!\cdots\!45}a^{3}+\frac{80\!\cdots\!72}{14\!\cdots\!89}a^{2}-\frac{71\!\cdots\!39}{37\!\cdots\!25}a-\frac{83\!\cdots\!91}{37\!\cdots\!25}$ (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: | \( 111003394.88977265 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 111003394.88977265 \cdot 1}{2\cdot\sqrt{725232993643865780590215168}}\cr\approx \mathstrut & 6.37406597492771 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 355687428096000 |
The 297 conjugacy class representatives for $S_{17}$ |
Character table for $S_{17}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $17$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.14.26.91 | $x^{14} + 2 x^{13} + 6 x^{12} + 4 x^{8} + 4 x^{7} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 4 x + 6$ | $14$ | $1$ | $26$ | 14T18 | $[2, 20/7, 20/7, 20/7]_{7}^{3}$ | |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.6.11.5 | $x^{6} + 18 x + 6$ | $6$ | $1$ | $11$ | $S_3^2$ | $[2, 5/2]_{2}^{2}$ | |
3.9.26.66 | $x^{9} + 27 x^{3} + 27 x^{2} + 27 x + 21$ | $9$ | $1$ | $26$ | $(C_3^3:C_3):C_2$ | $[3/2, 5/2, 8/3, 7/2]_{2}$ |