Normalized defining polynomial
\( x^{17} - 6 x^{15} - 2 x^{14} + 17 x^{13} + 13 x^{12} - 27 x^{11} - 34 x^{10} + 20 x^{9} + 47 x^{8} + \cdots - 1 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(4755937542258621701\) \(\medspace = 97\cdot 2927\cdot 2315407\cdot 7234597\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(12.55\) | 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))
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Galois root discriminant: | $97^{1/2}2927^{1/2}2315407^{1/2}7234597^{1/2}\approx 2180811211.971046$ | ||
Ramified primes: | \(97\), \(2927\), \(2315407\), \(7234597\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{47559\!\cdots\!21701}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1097}a^{16}-\frac{19}{1097}a^{15}+\frac{355}{1097}a^{14}-\frac{165}{1097}a^{13}-\frac{139}{1097}a^{12}+\frac{460}{1097}a^{11}+\frac{9}{1097}a^{10}-\frac{205}{1097}a^{9}-\frac{473}{1097}a^{8}+\frac{258}{1097}a^{7}-\frac{514}{1097}a^{6}-\frac{145}{1097}a^{5}+\frac{548}{1097}a^{4}-\frac{522}{1097}a^{3}+\frac{57}{1097}a^{2}+\frac{12}{1097}a-\frac{231}{1097}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $a$, $\frac{541}{1097}a^{16}-\frac{406}{1097}a^{15}-\frac{3211}{1097}a^{14}+\frac{1786}{1097}a^{13}+\frac{9270}{1097}a^{12}-\frac{2353}{1097}a^{11}-\frac{17071}{1097}a^{10}-\frac{108}{1097}a^{9}+\frac{20551}{1097}a^{8}+\frac{3550}{1097}a^{7}-\frac{18085}{1097}a^{6}-\frac{4946}{1097}a^{5}+\frac{12345}{1097}a^{4}+\frac{3915}{1097}a^{3}-\frac{4267}{1097}a^{2}-\frac{1187}{1097}a+\frac{1184}{1097}$, $\frac{868}{1097}a^{16}-\frac{37}{1097}a^{15}-\frac{4505}{1097}a^{14}-\frac{1707}{1097}a^{13}+\frac{10988}{1097}a^{12}+\frac{10942}{1097}a^{11}-\frac{14128}{1097}a^{10}-\frac{25457}{1097}a^{9}+\frac{4102}{1097}a^{8}+\frac{29775}{1097}a^{7}+\frac{10200}{1097}a^{6}-\frac{19451}{1097}a^{5}-\frac{14695}{1097}a^{4}+\frac{5450}{1097}a^{3}+\frac{8887}{1097}a^{2}+\frac{543}{1097}a-\frac{854}{1097}$, $\frac{638}{1097}a^{16}-\frac{55}{1097}a^{15}-\frac{3880}{1097}a^{14}-\frac{1055}{1097}a^{13}+\frac{11145}{1097}a^{12}+\frac{8260}{1097}a^{11}-\frac{18392}{1097}a^{10}-\frac{23284}{1097}a^{9}+\frac{15259}{1097}a^{8}+\frac{34061}{1097}a^{7}-\frac{2123}{1097}a^{6}-\frac{29981}{1097}a^{5}-\frac{7998}{1097}a^{4}+\frac{15810}{1097}a^{3}+\frac{8941}{1097}a^{2}-\frac{3314}{1097}a-\frac{2574}{1097}$, $\frac{765}{1097}a^{16}-\frac{274}{1097}a^{15}-\frac{4869}{1097}a^{14}-\frac{70}{1097}a^{13}+\frac{15432}{1097}a^{12}+\frac{6345}{1097}a^{11}-\frac{29316}{1097}a^{10}-\frac{22991}{1097}a^{9}+\frac{31978}{1097}a^{8}+\frac{40499}{1097}a^{7}-\frac{18036}{1097}a^{6}-\frac{40717}{1097}a^{5}+\frac{166}{1097}a^{4}+\frac{25209}{1097}a^{3}+\frac{7404}{1097}a^{2}-\frac{7275}{1097}a-\frac{3389}{1097}$, $\frac{219}{1097}a^{16}-\frac{870}{1097}a^{15}-\frac{1239}{1097}a^{14}+\frac{4454}{1097}a^{13}+\frac{4663}{1097}a^{12}-\frac{10057}{1097}a^{11}-\frac{13387}{1097}a^{10}+\frac{11052}{1097}a^{9}+\frac{23665}{1097}a^{8}-\frac{1639}{1097}a^{7}-\frac{23709}{1097}a^{6}-\frac{7621}{1097}a^{5}+\frac{12506}{1097}a^{4}+\frac{8546}{1097}a^{3}-\frac{1778}{1097}a^{2}-\frac{3954}{1097}a-\frac{1224}{1097}$, $\frac{325}{1097}a^{16}+\frac{407}{1097}a^{15}-\frac{2004}{1097}a^{14}-\frac{3163}{1097}a^{13}+\frac{5287}{1097}a^{12}+\frac{11278}{1097}a^{11}-\frac{5851}{1097}a^{10}-\frac{21648}{1097}a^{9}-\frac{2339}{1097}a^{8}+\frac{22418}{1097}a^{7}+\frac{11761}{1097}a^{6}-\frac{12021}{1097}a^{5}-\frac{11681}{1097}a^{4}+\frac{1482}{1097}a^{3}+\frac{5361}{1097}a^{2}+\frac{1706}{1097}a+\frac{618}{1097}$, $\frac{171}{1097}a^{16}+\frac{42}{1097}a^{15}-\frac{727}{1097}a^{14}-\frac{790}{1097}a^{13}+\frac{1462}{1097}a^{12}+\frac{2967}{1097}a^{11}-\frac{655}{1097}a^{10}-\frac{4339}{1097}a^{9}-\frac{2996}{1097}a^{8}+\frac{1335}{1097}a^{7}+\frac{4254}{1097}a^{6}+\frac{3727}{1097}a^{5}-\frac{1731}{1097}a^{4}-\frac{3696}{1097}a^{3}-\frac{1223}{1097}a^{2}+\frac{2052}{1097}a+\frac{1088}{1097}$, $\frac{187}{1097}a^{16}-\frac{262}{1097}a^{15}-\frac{1629}{1097}a^{14}+\frac{2055}{1097}a^{13}+\frac{5820}{1097}a^{12}-\frac{5031}{1097}a^{11}-\frac{13675}{1097}a^{10}+\frac{4448}{1097}a^{9}+\frac{22346}{1097}a^{8}+\frac{2172}{1097}a^{7}-\frac{21522}{1097}a^{6}-\frac{9563}{1097}a^{5}+\frac{11425}{1097}a^{4}+\frac{8795}{1097}a^{3}-\frac{2505}{1097}a^{2}-\frac{3241}{1097}a-\frac{1511}{1097}$, $\frac{16}{1097}a^{16}-\frac{304}{1097}a^{15}+\frac{195}{1097}a^{14}+\frac{1748}{1097}a^{13}-\frac{1127}{1097}a^{12}-\frac{4707}{1097}a^{11}+\frac{1241}{1097}a^{10}+\frac{8787}{1097}a^{9}+\frac{1208}{1097}a^{8}-\frac{11230}{1097}a^{7}-\frac{3836}{1097}a^{6}+\frac{8650}{1097}a^{5}+\frac{4380}{1097}a^{4}-\frac{5061}{1097}a^{3}-\frac{2379}{1097}a^{2}+\frac{2386}{1097}a+\frac{692}{1097}$ | 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: | \( 331.845168707 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{6}\cdot 331.845168707 \cdot 1}{2\cdot\sqrt{4755937542258621701}}\cr\approx \mathstrut & 0.149801658191 \end{aligned}\]
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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(97\) | 97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
97.3.0.1 | $x^{3} + 9 x + 92$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
97.12.0.1 | $x^{12} + 30 x^{7} + 59 x^{6} + 81 x^{5} + 86 x^{3} + 78 x^{2} + 94 x + 5$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(2927\) | $\Q_{2927}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(2315407\) | $\Q_{2315407}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2315407}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(7234597\) | $\Q_{7234597}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |