Normalized defining polynomial
\( x^{28} - 24 x^{24} - 160 x^{22} - 1656 x^{20} + 35040 x^{18} - 27616 x^{16} - 906528 x^{14} + \cdots - 1565568 \)
Invariants
Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-3206343011700717904956462085952501156673661196828672\) \(\medspace = -\,2^{77}\cdot 151^{13}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(69.10\) | 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: | $2^{11/4}151^{1/2}\approx 82.664865168866$ | ||
Ramified primes: | \(2\), \(151\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-302}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{12}a^{8}-\frac{1}{6}a^{6}-\frac{1}{6}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{12}a^{9}-\frac{1}{3}a$, $\frac{1}{12}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{12}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{24}a^{12}-\frac{1}{6}a^{4}$, $\frac{1}{24}a^{13}-\frac{1}{6}a^{5}$, $\frac{1}{72}a^{14}-\frac{1}{72}a^{12}-\frac{1}{36}a^{8}-\frac{1}{6}a^{6}+\frac{1}{9}a^{4}+\frac{2}{9}a^{2}$, $\frac{1}{72}a^{15}-\frac{1}{72}a^{13}-\frac{1}{36}a^{9}-\frac{2}{9}a^{5}-\frac{1}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{144}a^{16}+\frac{1}{72}a^{12}+\frac{1}{36}a^{10}+\frac{1}{36}a^{8}+\frac{2}{9}a^{6}-\frac{1}{6}a^{4}+\frac{4}{9}a^{2}$, $\frac{1}{144}a^{17}+\frac{1}{72}a^{13}+\frac{1}{36}a^{11}+\frac{1}{36}a^{9}+\frac{1}{18}a^{7}+\frac{1}{6}a^{5}-\frac{2}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{144}a^{18}+\frac{1}{36}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{144}a^{19}+\frac{1}{36}a^{11}-\frac{2}{9}a^{3}$, $\frac{1}{288}a^{20}+\frac{1}{72}a^{12}-\frac{1}{9}a^{4}$, $\frac{1}{864}a^{21}+\frac{1}{216}a^{15}-\frac{1}{72}a^{13}-\frac{1}{36}a^{11}+\frac{1}{54}a^{9}+\frac{1}{18}a^{7}+\frac{1}{6}a^{5}-\frac{1}{27}a^{3}-\frac{1}{3}a$, $\frac{1}{78624}a^{22}-\frac{5}{3744}a^{20}+\frac{3}{1456}a^{18}-\frac{103}{39312}a^{16}-\frac{5}{936}a^{14}-\frac{131}{6552}a^{12}+\frac{17}{756}a^{10}+\frac{67}{3276}a^{8}-\frac{17}{234}a^{6}+\frac{1021}{4914}a^{4}+\frac{44}{819}a^{2}+\frac{9}{91}$, $\frac{1}{78624}a^{23}-\frac{1}{5616}a^{21}+\frac{3}{1456}a^{19}-\frac{103}{39312}a^{17}-\frac{1}{1404}a^{15}+\frac{17}{2184}a^{13}-\frac{1}{189}a^{11}+\frac{383}{9828}a^{9}-\frac{2}{117}a^{7}+\frac{1021}{4914}a^{5}+\frac{41}{2457}a^{3}-\frac{64}{273}a$, $\frac{1}{1415232}a^{24}-\frac{1}{176904}a^{22}-\frac{5}{4536}a^{20}-\frac{1037}{353808}a^{18}-\frac{323}{353808}a^{16}-\frac{1}{273}a^{14}+\frac{1499}{176904}a^{12}+\frac{445}{88452}a^{10}+\frac{223}{14742}a^{8}+\frac{6401}{44226}a^{6}+\frac{1279}{22113}a^{4}-\frac{269}{819}a^{2}+\frac{100}{273}$, $\frac{1}{1415232}a^{25}-\frac{1}{176904}a^{23}+\frac{1}{18144}a^{21}-\frac{1037}{353808}a^{19}-\frac{323}{353808}a^{17}+\frac{19}{19656}a^{15}-\frac{479}{88452}a^{13}-\frac{503}{22113}a^{11}+\frac{248}{7371}a^{9}+\frac{1487}{44226}a^{7}+\frac{1279}{22113}a^{5}-\frac{79}{2457}a^{3}+\frac{100}{273}a$, $\frac{1}{10\!\cdots\!36}a^{26}-\frac{99\!\cdots\!25}{10\!\cdots\!36}a^{24}-\frac{82\!\cdots\!17}{25\!\cdots\!84}a^{22}-\frac{27\!\cdots\!15}{36\!\cdots\!12}a^{20}+\frac{28\!\cdots\!55}{36\!\cdots\!12}a^{18}-\frac{68\!\cdots\!05}{25\!\cdots\!84}a^{16}+\frac{79\!\cdots\!69}{12\!\cdots\!92}a^{14}+\frac{23\!\cdots\!87}{12\!\cdots\!92}a^{12}-\frac{79\!\cdots\!93}{63\!\cdots\!96}a^{10}-\frac{17\!\cdots\!45}{31\!\cdots\!98}a^{8}+\frac{30\!\cdots\!29}{24\!\cdots\!46}a^{6}-\frac{26\!\cdots\!71}{31\!\cdots\!98}a^{4}-\frac{13\!\cdots\!20}{64\!\cdots\!93}a^{2}-\frac{67\!\cdots\!26}{14\!\cdots\!83}$, $\frac{1}{10\!\cdots\!36}a^{27}-\frac{99\!\cdots\!25}{10\!\cdots\!36}a^{25}-\frac{82\!\cdots\!17}{25\!\cdots\!84}a^{23}+\frac{29\!\cdots\!61}{72\!\cdots\!24}a^{21}+\frac{28\!\cdots\!55}{36\!\cdots\!12}a^{19}-\frac{68\!\cdots\!05}{25\!\cdots\!84}a^{17}-\frac{37\!\cdots\!05}{12\!\cdots\!92}a^{15}+\frac{23\!\cdots\!87}{12\!\cdots\!92}a^{13}-\frac{45\!\cdots\!26}{15\!\cdots\!99}a^{11}+\frac{25\!\cdots\!95}{63\!\cdots\!96}a^{9}+\frac{34\!\cdots\!35}{24\!\cdots\!46}a^{7}+\frac{43\!\cdots\!73}{31\!\cdots\!98}a^{5}+\frac{75\!\cdots\!87}{19\!\cdots\!79}a^{3}-\frac{17\!\cdots\!65}{14\!\cdots\!83}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
not computed
Unit group
Rank: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | not computed | 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: | not computed | 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 $
Galois group
A solvable group of order 56 |
The 17 conjugacy class representatives for $D_{28}$ |
Character table for $D_{28}$ |
Intermediate fields
\(\Q(\sqrt{2}) \), 4.2.309248.4, 7.1.3442951.1, 14.2.24859454395438333952.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 28 sibling: | deg 28 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{14}$ | $28$ | ${\href{/padicField/7.2.0.1}{2} }^{13}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $28$ | ${\href{/padicField/13.2.0.1}{2} }^{14}$ | ${\href{/padicField/17.7.0.1}{7} }^{4}$ | $28$ | ${\href{/padicField/23.2.0.1}{2} }^{13}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $28$ | ${\href{/padicField/31.7.0.1}{7} }^{4}$ | $28$ | ${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $28$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{14}$ | $28$ |
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 $28$ | $4$ | $7$ | $77$ | |||
\(151\) | $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |