Normalized defining polynomial
\( x^{27} - 6 x^{26} + 31 x^{25} - 80 x^{24} + 117 x^{23} - 51 x^{22} - 243 x^{21} + 807 x^{20} - 1350 x^{19} + 966 x^{18} + 1935 x^{17} - 9345 x^{16} + 23103 x^{15} - 43548 x^{14} + 68751 x^{13} - 94575 x^{12} + 115269 x^{11} - 125268 x^{10} + 121785 x^{9} - 105567 x^{8} + 81027 x^{7} - 54357 x^{6} + 31122 x^{5} - 14664 x^{4} + 5393 x^{3} - 1350 x^{2} + 179 x - 1 \)
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: | \(-1543319746516623033280478216838436483146079\)\(\medspace = -\,1759^{13}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $36.52$ | 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: | $1759$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{12} - \frac{2}{9} a^{4}$, $\frac{1}{9} a^{21} + \frac{1}{9} a^{13} - \frac{2}{9} a^{5}$, $\frac{1}{27} a^{22} + \frac{1}{27} a^{21} - \frac{1}{27} a^{20} - \frac{1}{27} a^{18} - \frac{1}{27} a^{17} + \frac{1}{27} a^{16} + \frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{10} - \frac{1}{27} a^{9} + \frac{1}{27} a^{8} - \frac{2}{27} a^{6} - \frac{2}{27} a^{5} + \frac{2}{27} a^{4} + \frac{2}{27} a^{2} + \frac{2}{27} a - \frac{2}{27}$, $\frac{1}{27} a^{23} + \frac{1}{27} a^{21} + \frac{1}{27} a^{20} - \frac{1}{27} a^{19} - \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{1}{27} a^{15} + \frac{1}{27} a^{13} + \frac{1}{27} a^{12} - \frac{1}{27} a^{11} - \frac{1}{27} a^{9} - \frac{1}{27} a^{8} - \frac{2}{27} a^{7} - \frac{2}{27} a^{5} - \frac{2}{27} a^{4} + \frac{2}{27} a^{3} + \frac{2}{27} a + \frac{2}{27}$, $\frac{1}{27} a^{24} - \frac{1}{9} a^{8} + \frac{2}{27}$, $\frac{1}{202797} a^{25} + \frac{254}{22533} a^{24} + \frac{220}{67599} a^{23} + \frac{442}{28971} a^{22} - \frac{1472}{28971} a^{21} + \frac{7313}{202797} a^{20} + \frac{3403}{67599} a^{19} + \frac{10271}{202797} a^{18} + \frac{9236}{202797} a^{17} + \frac{235}{202797} a^{16} + \frac{3856}{67599} a^{15} + \frac{11230}{202797} a^{14} + \frac{766}{5481} a^{13} - \frac{30865}{202797} a^{12} + \frac{6691}{67599} a^{11} - \frac{8143}{202797} a^{10} + \frac{19178}{202797} a^{9} + \frac{31636}{202797} a^{8} + \frac{32230}{67599} a^{7} - \frac{2675}{28971} a^{6} - \frac{20306}{202797} a^{5} - \frac{72379}{202797} a^{4} - \frac{20714}{67599} a^{3} + \frac{5501}{28971} a^{2} + \frac{14401}{28971} a + \frac{17494}{202797}$, $\frac{1}{3401596152173459846238129} a^{26} + \frac{4307896242397707175}{3401596152173459846238129} a^{25} - \frac{58800317973104262888547}{3401596152173459846238129} a^{24} + \frac{696690136732004143805}{1133865384057819948746043} a^{23} - \frac{609424976164941145585}{161980769151117135535149} a^{22} - \frac{46066059511880976971}{1725822502371111033099} a^{21} + \frac{10230492152795716098671}{377955128019273316248681} a^{20} - \frac{44249908117598173524424}{1133865384057819948746043} a^{19} + \frac{4564622200880778041351}{161980769151117135535149} a^{18} - \frac{20032557056116754684054}{377955128019273316248681} a^{17} + \frac{9722229141917537957788}{377955128019273316248681} a^{16} + \frac{984162272193725786324}{7509042278528608932093} a^{15} + \frac{448355665696345932485}{7509042278528608932093} a^{14} - \frac{113341777598021175587}{3669467262323041905327} a^{13} - \frac{3831422452649316745766}{53993589717039045178383} a^{12} - \frac{2925478470162800351686}{1133865384057819948746043} a^{11} + \frac{135067254605921093428537}{1133865384057819948746043} a^{10} - \frac{152996910428730473766967}{1133865384057819948746043} a^{9} + \frac{86822969295656201363557}{1133865384057819948746043} a^{8} - \frac{205318045204383381101590}{1133865384057819948746043} a^{7} - \frac{347912314047464599446505}{1133865384057819948746043} a^{6} - \frac{178338945657990561998836}{377955128019273316248681} a^{5} + \frac{37701538126997227113758}{125985042673091105416227} a^{4} + \frac{227420428785178017308372}{1133865384057819948746043} a^{3} + \frac{91694467203047109886865}{485942307453351406605447} a^{2} + \frac{222647107579038322344767}{3401596152173459846238129} a + \frac{897274803829459144634932}{3401596152173459846238129}$
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: | \( 71844122831.24568 \) (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
3.1.1759.1, 9.1.9573337234561.1 |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ | $27$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | $27$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | $27$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
1759 | Data not computed |