Normalized defining polynomial
\( x^{28} - 3x - 1 \)
Invariants
Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-10144175740601324551903917792039296238848946961358419\) \(\medspace = -\,79\cdot 6399373412031876752263\cdot 20065603237699194720476206747\) | 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: | \(72.01\) | 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: | $79^{1/2}6399373412031876752263^{1/2}20065603237699194720476206747^{1/2}\approx 1.0071829893619791e+26$ | ||
Ramified primes: | \(79\), \(6399373412031876752263\), \(20065603237699194720476206747\) | 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: | $\Q(\sqrt{-10144\!\cdots\!58419}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $14$ | 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: | $a^{27}-3$, $a^{14}-a-1$, $a^{12}-a^{11}+a^{9}-a^{8}+a^{6}-a^{5}+a^{3}-a^{2}+1$, $a^{27}-a^{25}-a^{23}+a^{21}+2a^{19}+a^{18}-a^{17}+a^{16}-2a^{15}-2a^{14}+a^{13}-2a^{12}+a^{11}+3a^{10}-a^{9}+2a^{8}+a^{7}-3a^{6}+a^{5}-a^{4}-a^{3}+3a^{2}-2$, $a^{24}+a^{23}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{14}+2a^{13}-a^{11}-a^{10}-a^{9}-2a^{8}-a^{7}-2a^{6}-4a^{5}-3a^{4}-a^{3}-2a$, $a^{27}+a^{26}+a^{25}-a^{24}-2a^{23}+a^{21}+a^{20}-2a^{18}-2a^{17}+2a^{15}+a^{14}-a^{13}-3a^{12}-3a^{11}+2a^{10}+3a^{9}-a^{8}-4a^{7}-4a^{6}+a^{5}+5a^{4}+2a^{3}-5a^{2}-6a-3$, $2a^{26}+a^{25}+3a^{24}+3a^{22}+2a^{21}+a^{20}+4a^{19}+3a^{17}+5a^{16}-2a^{15}+7a^{14}+a^{13}+2a^{12}+7a^{11}-2a^{10}+8a^{9}+3a^{8}+a^{7}+8a^{6}+2a^{5}+5a^{4}+6a^{3}+2a^{2}+9a+4$, $3a^{27}-a^{26}+a^{24}-4a^{23}+a^{22}+a^{21}-a^{20}+5a^{19}-2a^{18}-2a^{17}+2a^{16}-4a^{15}+4a^{14}+3a^{13}-2a^{12}+4a^{11}-4a^{10}-3a^{9}+3a^{8}-2a^{7}+4a^{6}+3a^{5}-6a^{4}-a^{3}-3a^{2}-4a-2$, $2a^{27}+3a^{26}+3a^{25}-5a^{22}-2a^{21}-3a^{20}+3a^{18}+4a^{17}+5a^{16}+a^{15}-a^{14}-6a^{13}-4a^{12}-6a^{11}+a^{10}+4a^{9}+6a^{8}+9a^{7}+a^{6}-a^{5}-9a^{4}-8a^{3}-7a^{2}-1$, $a^{24}-a^{22}+a^{21}+2a^{20}-a^{19}-2a^{18}+2a^{17}+2a^{16}-a^{15}+a^{13}+a^{12}+2a^{9}+a^{8}-2a^{7}-a^{6}+4a^{5}+2a^{4}-3a^{3}+3a+2$, $3a^{25}-3a^{24}-a^{23}+3a^{22}-a^{20}-4a^{19}+6a^{18}-4a^{16}+3a^{14}+2a^{13}-7a^{12}+2a^{11}+5a^{10}-a^{9}-5a^{8}+a^{7}+5a^{6}-2a^{5}-5a^{4}+3a^{3}+6a^{2}-5a-3$, $7a^{27}-3a^{26}-7a^{25}+8a^{24}-a^{23}-5a^{22}+11a^{21}-a^{20}-8a^{19}+9a^{18}-4a^{17}-12a^{16}+10a^{15}-a^{14}-9a^{13}+14a^{12}+a^{11}-12a^{10}+11a^{9}-5a^{8}-17a^{7}+13a^{6}+a^{5}-12a^{4}+22a^{3}+5a^{2}-17a-6$, $a^{26}+3a^{25}+a^{24}+a^{22}-a^{21}-2a^{20}-2a^{18}-2a^{17}-a^{16}-a^{15}-a^{14}+3a^{13}+2a^{12}+3a^{11}+4a^{10}+a^{9}-a^{8}+a^{7}-4a^{6}-4a^{5}-2a^{4}-5a^{3}-5a^{2}+2a+2$, $3a^{27}+3a^{26}-6a^{25}+5a^{24}-2a^{23}-a^{22}+4a^{21}-6a^{20}+5a^{19}+a^{18}-8a^{17}+10a^{16}-4a^{15}-6a^{14}+11a^{13}-6a^{12}-2a^{11}+5a^{10}-6a^{9}+6a^{8}-2a^{7}-6a^{6}+11a^{5}-10a^{4}+14a^{2}-17a-6$ (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: | \( 1126285775019767.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{13}\cdot 1126285775019767.5 \cdot 1}{2\cdot\sqrt{10144175740601324551903917792039296238848946961358419}}\cr\approx \mathstrut & 0.531996164388714 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 304888344611713860501504000000 |
The 3718 conjugacy class representatives for $S_{28}$ |
Character table for $S_{28}$ |
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 | $28$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/5.9.0.1}{9} }$ | $16{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$ | $22{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/37.6.0.1}{6} }$ | $17{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | $25{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | $26{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(79\) | $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
79.2.1.1 | $x^{2} + 237$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
79.6.0.1 | $x^{6} + 19 x^{3} + 28 x^{2} + 68 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
79.18.0.1 | $x^{18} + x^{2} - 3 x + 35$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(639\!\cdots\!263\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(200\!\cdots\!747\) | $\Q_{20\!\cdots\!47}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ |