Normalized defining polynomial
\( x^{18} - 4x^{16} - 16x^{14} + 69x^{12} + 3x^{10} - 232x^{8} + 290x^{6} - 133x^{4} + 22x^{2} - 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[14, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(104192253035948620552339456\) \(\medspace = 2^{18}\cdot 7^{12}\cdot 169457^{2}\) | 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: | \(27.89\) | 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: | not computed | ||
Ramified primes: | \(2\), \(7\), \(169457\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{161657}a^{16}-\frac{299}{1163}a^{14}+\frac{7073}{161657}a^{12}-\frac{40166}{161657}a^{10}+\frac{69940}{161657}a^{8}-\frac{65609}{161657}a^{6}+\frac{6541}{161657}a^{4}-\frac{79053}{161657}a^{2}+\frac{11989}{161657}$, $\frac{1}{161657}a^{17}-\frac{299}{1163}a^{15}+\frac{7073}{161657}a^{13}-\frac{40166}{161657}a^{11}+\frac{69940}{161657}a^{9}-\frac{65609}{161657}a^{7}+\frac{6541}{161657}a^{5}-\frac{79053}{161657}a^{3}+\frac{11989}{161657}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $15$ | 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: | $\frac{177479}{161657}a^{16}-\frac{4346}{1163}a^{14}-\frac{3190778}{161657}a^{12}+\frac{10313263}{161657}a^{10}+\frac{6514795}{161657}a^{8}-\frac{36762140}{161657}a^{6}+\frac{29937767}{161657}a^{4}-\frac{7795893}{161657}a^{2}+\frac{712925}{161657}$, $\frac{16681}{161657}a^{16}-\frac{675}{1163}a^{14}-\frac{186554}{161657}a^{12}+\frac{1675789}{161657}a^{10}-\frac{1302685}{161657}a^{8}-\frac{5502177}{161657}a^{6}+\frac{9853023}{161657}a^{4}-\frac{4411683}{161657}a^{2}+\frac{342114}{161657}$, $a^{17}-4a^{15}-16a^{13}+69a^{11}+3a^{9}-232a^{7}+290a^{5}-133a^{3}+22a$, $\frac{31675}{161657}a^{17}-\frac{516}{1163}a^{15}-\frac{665955}{161657}a^{13}+\frac{1114139}{161657}a^{11}+\frac{2750141}{161657}a^{9}-\frac{3944108}{161657}a^{7}-\frac{543070}{161657}a^{5}+\frac{63155}{161657}a^{3}+\frac{665910}{161657}a$, $\frac{373789}{161657}a^{17}-\frac{10241}{1163}a^{15}-\frac{6233604}{161657}a^{13}+\frac{24533451}{161657}a^{11}+\frac{5452272}{161657}a^{9}-\frac{84617241}{161657}a^{7}+\frac{93167813}{161657}a^{5}-\frac{35584984}{161657}a^{3}+\frac{3780735}{161657}a$, $\frac{241815}{161657}a^{17}-\frac{5953}{1163}a^{15}-\frac{4338304}{161657}a^{13}+\frac{14160400}{161657}a^{11}+\frac{8715238}{161657}a^{9}-\frac{50820996}{161657}a^{7}+\frac{41605676}{161657}a^{5}-\frac{8990423}{161657}a^{3}-\frac{359917}{161657}a$, $\frac{212132}{161657}a^{17}-\frac{5589}{1163}a^{15}-\frac{3647092}{161657}a^{13}+\frac{13379118}{161657}a^{11}+\frac{4967301}{161657}a^{9}-\frac{47112817}{161657}a^{7}+\frac{46287283}{161657}a^{5}-\frac{14084603}{161657}a^{3}+\frac{224281}{161657}a$, $\frac{113301}{161657}a^{17}-\frac{3461}{1163}a^{15}-\frac{1734003}{161657}a^{13}+\frac{8364405}{161657}a^{11}-\frac{962485}{161657}a^{9}-\frac{28058139}{161657}a^{7}+\frac{37570577}{161657}a^{5}-\frac{17151853}{161657}a^{3}+\frac{2548430}{161657}a$, $\frac{31675}{161657}a^{17}-\frac{516}{1163}a^{15}-\frac{665955}{161657}a^{13}+\frac{1114139}{161657}a^{11}+\frac{2750141}{161657}a^{9}-\frac{3944108}{161657}a^{7}-\frac{543070}{161657}a^{5}+\frac{63155}{161657}a^{3}+\frac{665910}{161657}a-1$, $\frac{241815}{161657}a^{17}-\frac{5953}{1163}a^{15}-\frac{4338304}{161657}a^{13}+\frac{14160400}{161657}a^{11}+\frac{8715238}{161657}a^{9}-\frac{50820996}{161657}a^{7}+\frac{41605676}{161657}a^{5}-\frac{8990423}{161657}a^{3}-\frac{359917}{161657}a+1$, $a+1$, $\frac{29683}{161657}a^{17}+\frac{177479}{161657}a^{16}-\frac{364}{1163}a^{15}-\frac{4346}{1163}a^{14}-\frac{691212}{161657}a^{13}-\frac{3190778}{161657}a^{12}+\frac{781282}{161657}a^{11}+\frac{10313263}{161657}a^{10}+\frac{3747937}{161657}a^{9}+\frac{6514795}{161657}a^{8}-\frac{3708179}{161657}a^{7}-\frac{36762140}{161657}a^{6}-\frac{4681607}{161657}a^{5}+\frac{29937767}{161657}a^{4}+\frac{5094180}{161657}a^{3}-\frac{7795893}{161657}a^{2}-\frac{584198}{161657}a+\frac{551268}{161657}$, $\frac{238624}{161657}a^{17}-\frac{16681}{161657}a^{16}-\frac{6667}{1163}a^{15}+\frac{675}{1163}a^{14}-\frac{3952953}{161657}a^{13}+\frac{186554}{161657}a^{12}+\frac{16075989}{161657}a^{11}-\frac{1675789}{161657}a^{10}+\frac{3127020}{161657}a^{9}+\frac{1302685}{161657}a^{8}-\frac{56143173}{161657}a^{7}+\frac{5502177}{161657}a^{6}+\frac{60824281}{161657}a^{5}-\frac{9853023}{161657}a^{4}-\frac{20556524}{161657}a^{3}+\frac{4411683}{161657}a^{2}+\frac{1312463}{161657}a-\frac{180457}{161657}$, $\frac{225835}{161657}a^{17}-\frac{177479}{161657}a^{16}-\frac{5537}{1163}a^{15}+\frac{4346}{1163}a^{14}-\frac{4043287}{161657}a^{13}+\frac{3190778}{161657}a^{12}+\frac{13103191}{161657}a^{11}-\frac{10313263}{161657}a^{10}+\frac{7962251}{161657}a^{9}-\frac{6514795}{161657}a^{8}-\frac{46208425}{161657}a^{7}+\frac{36762140}{161657}a^{6}+\frac{39247720}{161657}a^{5}-\frac{29937767}{161657}a^{4}-\frac{12144421}{161657}a^{3}+\frac{7795893}{161657}a^{2}+\frac{1720949}{161657}a-\frac{712925}{161657}$, $\frac{31675}{161657}a^{17}+\frac{273490}{161657}a^{16}-\frac{516}{1163}a^{15}-\frac{6469}{1163}a^{14}-\frac{665955}{161657}a^{13}-\frac{5004259}{161657}a^{12}+\frac{1114139}{161657}a^{11}+\frac{15274539}{161657}a^{10}+\frac{2750141}{161657}a^{9}+\frac{11465379}{161657}a^{8}-\frac{3944108}{161657}a^{7}-\frac{54765104}{161657}a^{6}-\frac{543070}{161657}a^{5}+\frac{41062606}{161657}a^{4}+\frac{63155}{161657}a^{3}-\frac{8927268}{161657}a^{2}+\frac{665910}{161657}a+\frac{305993}{161657}$ (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: | \( 5324209.8837 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 5324209.8837 \cdot 1}{2\cdot\sqrt{104192253035948620552339456}}\cr\approx \mathstrut & 0.16868914345 \end{aligned}\] (assuming GRH)
Galois group
$S_4^3.A_4$ (as 18T838):
A solvable group of order 165888 |
The 180 conjugacy class representatives for $S_4^3.A_4$ |
Character table for $S_4^3.A_4$ |
Intermediate fields
\(\Q(\zeta_{7})^+\), 9.9.19936446593.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 18.6.104192253035948620552339456.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.18.18.82 | $x^{18} + 8 x^{17} + 26 x^{16} + 856 x^{15} + 10168 x^{14} + 50448 x^{13} + 118768 x^{12} + 93472 x^{11} - 126720 x^{10} - 246720 x^{9} - 257792 x^{8} - 1251200 x^{7} - 2384768 x^{6} + 1268480 x^{5} + 9044224 x^{4} + 11077120 x^{3} - 3846400 x^{2} - 15465472 x - 13969920$ | $2$ | $9$ | $18$ | 18T368 | $[2, 2, 2, 2, 2, 2, 2, 2]^{9}$ |
\(7\) | 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
\(169457\) | $\Q_{169457}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{169457}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |