Normalized defining polynomial
\( x^{18} - x^{17} + 4 x^{15} - 5 x^{14} + x^{13} + 6 x^{12} - 9 x^{11} + 7 x^{10} + x^{9} - 12 x^{8} + 11 x^{7} - 2 x^{6} - 5 x^{5} + 6 x^{4} - x^{3} - x^{2} - x + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-13950087665617805207\) \(\medspace = -\,11^{4}\cdot 23^{11}\) | 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: | \(11.58\) | 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: | $11^{1/2}23^{3/4}\approx 34.833107461122644$ | ||
Ramified primes: | \(11\), \(23\) | 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: | \(\Q(\sqrt{-23}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{16867}a^{17}-\frac{6775}{16867}a^{16}-\frac{1257}{16867}a^{15}-\frac{2913}{16867}a^{14}-\frac{1733}{16867}a^{13}-\frac{89}{16867}a^{12}-\frac{4320}{16867}a^{11}-\frac{574}{16867}a^{10}-\frac{7994}{16867}a^{9}+\frac{8287}{16867}a^{8}-\frac{2774}{16867}a^{7}+\frac{1249}{16867}a^{6}+\frac{6506}{16867}a^{5}+\frac{1822}{16867}a^{4}+\frac{4422}{16867}a^{3}+\frac{1163}{16867}a^{2}-\frac{1274}{16867}a-\frac{5829}{16867}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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)
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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: | $\frac{17410}{16867}a^{17}-\frac{1819}{16867}a^{16}+\frac{8996}{16867}a^{15}+\frac{54340}{16867}a^{14}-\frac{13334}{16867}a^{13}+\frac{19141}{16867}a^{12}+\frac{49354}{16867}a^{11}-\frac{24943}{16867}a^{10}+\frac{61545}{16867}a^{9}+\frac{13219}{16867}a^{8}-\frac{55720}{16867}a^{7}-\frac{13340}{16867}a^{6}-\frac{9312}{16867}a^{5}-\frac{5807}{16867}a^{4}-\frac{27702}{16867}a^{3}+\frac{41164}{16867}a^{2}-\frac{235}{16867}a-\frac{11018}{16867}$, $\frac{5790}{16867}a^{17}+\frac{5392}{16867}a^{16}-\frac{8353}{16867}a^{15}+\frac{34464}{16867}a^{14}+\frac{1795}{16867}a^{13}-\frac{26167}{16867}a^{12}+\frac{68429}{16867}a^{11}-\frac{34395}{16867}a^{10}-\frac{2212}{16867}a^{9}+\frac{79450}{16867}a^{8}-\frac{105278}{16867}a^{7}+\frac{29501}{16867}a^{6}+\frac{22596}{16867}a^{5}-\frac{76830}{16867}a^{4}+\frac{33008}{16867}a^{3}-\frac{13030}{16867}a^{2}+\frac{11286}{16867}a+\frac{957}{16867}$, $\frac{15616}{16867}a^{17}-\frac{8576}{16867}a^{16}+\frac{3876}{16867}a^{15}+\frac{51492}{16867}a^{14}-\frac{41594}{16867}a^{13}+\frac{10137}{16867}a^{12}+\frac{57481}{16867}a^{11}-\frac{74675}{16867}a^{10}+\frac{65831}{16867}a^{9}+\frac{6168}{16867}a^{8}-\frac{122397}{16867}a^{7}+\frac{56733}{16867}a^{6}-\frac{9112}{16867}a^{5}-\frac{52878}{16867}a^{4}+\frac{34188}{16867}a^{3}+\frac{12516}{16867}a^{2}-\frac{8591}{16867}a+\frac{5535}{16867}$, $\frac{9857}{16867}a^{17}-\frac{4722}{16867}a^{16}+\frac{6996}{16867}a^{15}+\frac{27927}{16867}a^{14}-\frac{12777}{16867}a^{13}+\frac{16678}{16867}a^{12}+\frac{23802}{16867}a^{11}-\frac{7473}{16867}a^{10}+\frac{39500}{16867}a^{9}-\frac{1922}{16867}a^{8}-\frac{18778}{16867}a^{7}-\frac{1517}{16867}a^{6}+\frac{18175}{16867}a^{5}-\frac{3901}{16867}a^{4}-\frac{30408}{16867}a^{3}+\frac{27865}{16867}a^{2}-\frac{8770}{16867}a-\frac{7451}{16867}$, $\frac{2247}{16867}a^{17}+\frac{7476}{16867}a^{16}-\frac{7690}{16867}a^{15}+\frac{15752}{16867}a^{14}+\frac{19093}{16867}a^{13}-\frac{31313}{16867}a^{12}+\frac{42086}{16867}a^{11}+\frac{8981}{16867}a^{10}-\frac{32897}{16867}a^{9}+\frac{84056}{16867}a^{8}-\frac{42989}{16867}a^{7}-\frac{27153}{16867}a^{6}+\frac{62761}{16867}a^{5}-\frac{55248}{16867}a^{4}+\frac{18438}{16867}a^{3}+\frac{32610}{16867}a^{2}-\frac{12155}{16867}a-\frac{8971}{16867}$, $\frac{9221}{16867}a^{17}+\frac{3093}{16867}a^{16}-\frac{3168}{16867}a^{15}+\frac{42092}{16867}a^{14}-\frac{6944}{16867}a^{13}-\frac{11053}{16867}a^{12}+\frac{72602}{16867}a^{11}-\frac{47217}{16867}a^{10}+\frac{29850}{16867}a^{9}+\frac{74385}{16867}a^{8}-\frac{109884}{16867}a^{7}+\frac{47469}{16867}a^{6}+\frac{12774}{16867}a^{5}-\frac{66338}{16867}a^{4}+\frac{58324}{16867}a^{3}+\frac{13478}{16867}a^{2}+\frac{8745}{16867}a-\frac{10947}{16867}$, $\frac{9801}{16867}a^{17}+\frac{3604}{16867}a^{16}-\frac{6947}{16867}a^{15}+\frac{39252}{16867}a^{14}-\frac{64}{16867}a^{13}-\frac{28939}{16867}a^{12}+\frac{63318}{16867}a^{11}-\frac{25930}{16867}a^{10}-\frac{1979}{16867}a^{9}+\frac{73750}{16867}a^{8}-\frac{99572}{16867}a^{7}-\frac{3993}{16867}a^{6}+\frac{58647}{16867}a^{5}-\frac{55332}{16867}a^{4}+\frac{25566}{16867}a^{3}+\frac{30205}{16867}a^{2}-\frac{21761}{16867}a-\frac{1500}{16867}$, $\frac{17410}{16867}a^{17}-\frac{1819}{16867}a^{16}+\frac{8996}{16867}a^{15}+\frac{54340}{16867}a^{14}-\frac{13334}{16867}a^{13}+\frac{19141}{16867}a^{12}+\frac{49354}{16867}a^{11}-\frac{24943}{16867}a^{10}+\frac{61545}{16867}a^{9}+\frac{13219}{16867}a^{8}-\frac{55720}{16867}a^{7}-\frac{13340}{16867}a^{6}-\frac{9312}{16867}a^{5}-\frac{5807}{16867}a^{4}-\frac{27702}{16867}a^{3}+\frac{41164}{16867}a^{2}+\frac{16632}{16867}a-\frac{11018}{16867}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 91.5286994248 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 91.5286994248 \cdot 1}{2\cdot\sqrt{13950087665617805207}}\cr\approx \mathstrut & 0.187006904927 \end{aligned}\]
Galois group
$C_3^3:S_4$ (as 18T217):
A solvable group of order 648 |
The 14 conjugacy class representatives for $C_3^3:S_4$ |
Character table for $C_3^3:S_4$ |
Intermediate fields
\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.33860761.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 9.1.33860761.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.4.2.2 | $x^{4} - 77 x^{2} + 242$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
11.4.2.2 | $x^{4} - 77 x^{2} + 242$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.3.1 | $x^{4} + 23$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
23.4.3.1 | $x^{4} + 23$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |