Normalized defining polynomial
\( x^{27} - 4 x^{26} + 14 x^{25} - 10 x^{24} + 37 x^{22} - 3 x^{21} - x^{20} + 139 x^{19} - 26 x^{18} + 76 x^{17} + 144 x^{16} - 117 x^{15} - 101 x^{14} + 12 x^{13} - 340 x^{12} + x^{11} + 205 x^{10} + 307 x^{9} + 774 x^{8} + 909 x^{7} + 892 x^{6} + 864 x^{5} + 521 x^{4} + 327 x^{3} + 86 x^{2} + 71 x + 1 \)
Invariants
Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[1, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(-14905779350658261917360296447434047234191\)\(\medspace = -\,1231^{13}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $30.75$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $1231$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $1$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{12} - \frac{2}{9} a^{4}$, $\frac{1}{27} a^{21} - \frac{1}{27} a^{19} + \frac{1}{27} a^{18} + \frac{1}{27} a^{17} + \frac{1}{27} a^{16} - \frac{1}{9} a^{15} - \frac{2}{27} a^{13} - \frac{1}{9} a^{12} + \frac{2}{27} a^{11} + \frac{1}{27} a^{10} + \frac{4}{27} a^{9} + \frac{4}{27} a^{8} - \frac{2}{9} a^{7} + \frac{1}{3} a^{6} - \frac{8}{27} a^{5} + \frac{4}{9} a^{4} - \frac{10}{27} a^{3} + \frac{7}{27} a^{2} + \frac{13}{27} a + \frac{4}{27}$, $\frac{1}{27} a^{22} - \frac{1}{27} a^{20} + \frac{1}{27} a^{19} + \frac{1}{27} a^{18} + \frac{1}{27} a^{17} - \frac{2}{27} a^{14} - \frac{1}{9} a^{13} + \frac{2}{27} a^{12} + \frac{1}{27} a^{11} + \frac{4}{27} a^{10} + \frac{4}{27} a^{9} - \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{8}{27} a^{6} + \frac{4}{9} a^{5} - \frac{10}{27} a^{4} + \frac{7}{27} a^{3} + \frac{13}{27} a^{2} + \frac{4}{27} a - \frac{2}{9}$, $\frac{1}{81} a^{23} + \frac{1}{81} a^{22} + \frac{1}{81} a^{21} + \frac{1}{27} a^{20} + \frac{4}{81} a^{18} + \frac{2}{81} a^{16} - \frac{8}{81} a^{15} - \frac{5}{81} a^{14} + \frac{4}{81} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{10}{81} a^{10} - \frac{1}{27} a^{9} + \frac{5}{81} a^{8} - \frac{38}{81} a^{7} - \frac{5}{81} a^{6} + \frac{4}{81} a^{5} + \frac{11}{27} a^{4} - \frac{4}{9} a^{3} - \frac{23}{81} a^{2} + \frac{4}{27} a + \frac{38}{81}$, $\frac{1}{135999} a^{24} + \frac{206}{135999} a^{23} + \frac{1538}{135999} a^{22} - \frac{1697}{135999} a^{21} - \frac{2246}{45333} a^{20} + \frac{3988}{135999} a^{19} - \frac{3137}{135999} a^{18} - \frac{742}{135999} a^{17} - \frac{121}{15111} a^{16} + \frac{128}{135999} a^{15} + \frac{11759}{135999} a^{14} - \frac{6080}{135999} a^{13} - \frac{1181}{15111} a^{12} - \frac{16958}{135999} a^{11} + \frac{2437}{135999} a^{10} + \frac{13640}{135999} a^{9} + \frac{53}{15111} a^{8} - \frac{51472}{135999} a^{7} - \frac{157}{1863} a^{6} - \frac{485}{135999} a^{5} - \frac{14119}{45333} a^{4} + \frac{29305}{135999} a^{3} + \frac{5695}{135999} a^{2} - \frac{52588}{135999} a - \frac{56035}{135999}$, $\frac{1}{21623841} a^{25} + \frac{58}{21623841} a^{24} + \frac{93617}{21623841} a^{23} - \frac{41273}{21623841} a^{22} + \frac{205801}{21623841} a^{21} - \frac{509888}{21623841} a^{20} + \frac{70070}{2402649} a^{19} - \frac{217667}{7207947} a^{18} + \frac{647686}{21623841} a^{17} + \frac{2937}{88987} a^{16} - \frac{3541480}{21623841} a^{15} + \frac{2405755}{21623841} a^{14} + \frac{100081}{21623841} a^{13} - \frac{84328}{940167} a^{12} + \frac{16042}{313389} a^{11} + \frac{835195}{7207947} a^{10} - \frac{5197}{407997} a^{9} - \frac{701096}{7207947} a^{8} - \frac{2521333}{21623841} a^{7} + \frac{9580327}{21623841} a^{6} - \frac{7559657}{21623841} a^{5} - \frac{9563372}{21623841} a^{4} - \frac{2451215}{7207947} a^{3} + \frac{2971090}{7207947} a^{2} - \frac{1035866}{7207947} a - \frac{9106297}{21623841}$, $\frac{1}{717044643038151} a^{26} + \frac{15099116}{717044643038151} a^{25} - \frac{445493129}{239014881012717} a^{24} - \frac{525974495980}{239014881012717} a^{23} - \frac{6273715611433}{717044643038151} a^{22} + \frac{7554204232715}{717044643038151} a^{21} + \frac{38591987442955}{717044643038151} a^{20} + \frac{32820573286}{611291255787} a^{19} - \frac{37328321562947}{717044643038151} a^{18} + \frac{357275968862}{8639092084797} a^{17} - \frac{376548944054}{42179096649303} a^{16} - \frac{1200727106698}{26557209001413} a^{15} - \frac{1565854272901}{9822529356687} a^{14} - \frac{71411258304379}{717044643038151} a^{13} + \frac{53009882666989}{717044643038151} a^{12} + \frac{12441069640637}{79671627004239} a^{11} + \frac{9892065981073}{717044643038151} a^{10} + \frac{28139744111119}{717044643038151} a^{9} + \frac{5385579118484}{717044643038151} a^{8} - \frac{10536512001476}{239014881012717} a^{7} - \frac{276749630863258}{717044643038151} a^{6} - \frac{98538605571637}{717044643038151} a^{5} + \frac{208887455610868}{717044643038151} a^{4} - \frac{45066103184386}{239014881012717} a^{3} + \frac{87820955548717}{239014881012717} a^{2} - \frac{344841463323733}{717044643038151} a + \frac{42099399472001}{717044643038151}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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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);
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Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 5105816638.575763 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
A solvable group of order 54 |
The 15 conjugacy class representatives for $D_{27}$ |
Character table for $D_{27}$ |
Intermediate fields
3.1.1231.1, 9.1.2296318960321.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $27$ | ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | $27$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ | $27$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $27$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | $27$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
1231 | Data not computed |