Normalized defining polynomial
\( x^{28} - 2x + 2 \)
Invariants
Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(4329685414058356179785542334930922683994094960640\) \(\medspace = 2^{28}\cdot 5\cdot 32\!\cdots\!13\) | 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: | \(54.58\) | 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))
| |
Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(5\), \(32258\!\cdots\!02713\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{16129\!\cdots\!13565}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
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);
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: | $a-1$, $a^{4}-a^{2}+1$, $a^{19}+a^{10}+a-1$, $a^{21}+a^{14}+a-1$, $a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{11}+a^{8}+a^{5}+a^{2}-3$, $a^{24}+a^{23}-a^{17}-a^{16}+a^{10}+a^{9}-a^{5}-a^{4}+1$, $a^{27}-a^{23}+a^{21}+a^{19}+a^{18}-a^{16}-a^{14}+a^{12}+a^{10}+a^{9}-2a^{8}-a^{5}+2a^{3}-1$, $3a^{27}+2a^{26}-a^{25}-3a^{24}-2a^{23}+a^{22}+3a^{21}+3a^{20}-3a^{18}-4a^{17}-a^{16}+2a^{15}+4a^{14}+2a^{13}-a^{12}-4a^{11}-2a^{10}+a^{9}+4a^{8}+3a^{7}-5a^{5}-5a^{4}-a^{3}+4a^{2}+5a-3$, $a^{27}-a^{25}+a^{23}+2a^{21}+a^{20}-a^{19}+2a^{15}+2a^{14}-a^{13}-a^{11}-a^{10}+3a^{9}+2a^{8}-a^{7}-a^{5}-2a^{4}+3a^{3}+2a^{2}-a-1$, $2a^{27}+a^{26}+a^{25}+a^{24}-a^{22}-a^{21}-a^{20}-a^{19}+a^{17}-a^{15}+a^{13}-a^{11}-a^{10}-a^{9}-a^{8}-a^{7}-a^{6}-a^{5}-a^{4}+2a^{2}+2a-5$, $a^{27}-a^{26}+2a^{25}+a^{24}-a^{23}+a^{22}-2a^{21}-a^{20}+2a^{19}-a^{18}+2a^{17}-3a^{15}+2a^{14}-a^{13}+3a^{11}-3a^{10}+a^{9}+a^{8}-3a^{7}+4a^{6}-a^{5}-a^{4}+3a^{3}-5a^{2}+a+1$, $2a^{27}+a^{26}+2a^{25}+a^{24}+a^{23}+2a^{22}-a^{21}+2a^{20}-2a^{19}+a^{18}-a^{17}+a^{16}+2a^{13}-2a^{12}+3a^{11}-2a^{10}+2a^{9}-2a^{8}+a^{7}-2a^{6}-a^{5}+2a^{4}-4a^{3}+5a^{2}-4a-1$, $4a^{27}-5a^{25}-a^{24}+4a^{23}+3a^{22}-3a^{21}-4a^{20}+3a^{19}+4a^{18}-a^{17}-5a^{16}+6a^{14}+a^{13}-5a^{12}-3a^{11}+5a^{10}+5a^{9}-5a^{8}-6a^{7}+3a^{6}+8a^{5}-10a^{3}-a^{2}+9a-3$ (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)]
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Regulator: | \( 19117757369089.4 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 19117757369089.4 \cdot 1}{2\cdot\sqrt{4329685414058356179785542334930922683994094960640}}\cr\approx \mathstrut & 0.686589149173923 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 304888344611713860501504000000 |
The 3718 conjugacy class representatives for $S_{28}$ are not computed |
Character table for $S_{28}$ is not computed |
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 | R | ${\href{/padicField/3.12.0.1}{12} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | $22{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | $19{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{3}{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $27{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $28$ | $28$ | $1$ | $28$ | |||
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
5.22.0.1 | $x^{22} + x^{12} + 3 x^{11} + 4 x^{9} + 3 x^{8} + 2 x^{6} + 2 x^{5} + 4 x^{3} + 3 x^{2} + 3 x + 2$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | |
\(322\!\cdots\!713\) | $\Q_{32\!\cdots\!13}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |