Normalized defining polynomial
\( x^{28} - 15 x^{26} - 390 x^{24} + 2850 x^{22} + 49825 x^{20} + 260250 x^{18} + 737875 x^{16} + \cdots - 11796875 \)
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: | \(-2715878171109088684095776889728000000000000000000000\) \(\medspace = -\,2^{28}\cdot 5^{21}\cdot 151^{13}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(68.70\) | 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: | $2\cdot 5^{3/4}151^{1/2}\approx 82.17618445784255$ | ||
Ramified primes: | \(2\), \(5\), \(151\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-755}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{75}a^{8}-\frac{1}{3}$, $\frac{1}{75}a^{9}-\frac{1}{3}a$, $\frac{1}{75}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{75}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{375}a^{12}-\frac{1}{15}a^{4}$, $\frac{1}{375}a^{13}-\frac{1}{15}a^{5}$, $\frac{1}{375}a^{14}-\frac{1}{15}a^{6}$, $\frac{1}{375}a^{15}-\frac{1}{15}a^{7}$, $\frac{1}{39375}a^{16}+\frac{1}{875}a^{14}+\frac{1}{2625}a^{12}-\frac{1}{315}a^{8}+\frac{8}{105}a^{4}-\frac{1}{7}a^{2}-\frac{23}{63}$, $\frac{1}{39375}a^{17}+\frac{1}{875}a^{15}+\frac{1}{2625}a^{13}-\frac{1}{315}a^{9}+\frac{8}{105}a^{5}-\frac{1}{7}a^{3}-\frac{23}{63}a$, $\frac{1}{39375}a^{18}-\frac{1}{2625}a^{14}-\frac{1}{875}a^{12}-\frac{1}{315}a^{10}-\frac{2}{525}a^{8}+\frac{1}{105}a^{6}+\frac{1}{35}a^{4}+\frac{4}{63}a^{2}+\frac{2}{21}$, $\frac{1}{39375}a^{19}-\frac{1}{2625}a^{15}-\frac{1}{875}a^{13}-\frac{1}{315}a^{11}-\frac{2}{525}a^{9}+\frac{1}{105}a^{7}+\frac{1}{35}a^{5}+\frac{4}{63}a^{3}+\frac{2}{21}a$, $\frac{1}{590625}a^{20}-\frac{1}{118125}a^{16}-\frac{1}{2625}a^{14}+\frac{22}{23625}a^{12}-\frac{2}{525}a^{10}+\frac{2}{675}a^{8}+\frac{2}{21}a^{6}-\frac{32}{945}a^{4}-\frac{1}{3}a^{2}-\frac{4}{189}$, $\frac{1}{590625}a^{21}-\frac{1}{118125}a^{17}-\frac{1}{2625}a^{15}+\frac{22}{23625}a^{13}-\frac{2}{525}a^{11}+\frac{2}{675}a^{9}+\frac{2}{21}a^{7}-\frac{32}{945}a^{5}-\frac{1}{3}a^{3}-\frac{4}{189}a$, $\frac{1}{590625}a^{22}-\frac{1}{118125}a^{18}-\frac{2}{3375}a^{14}-\frac{2}{2625}a^{12}+\frac{2}{675}a^{10}-\frac{1}{175}a^{8}+\frac{31}{945}a^{6}+\frac{8}{105}a^{4}-\frac{31}{189}a^{2}-\frac{1}{7}$, $\frac{1}{590625}a^{23}-\frac{1}{118125}a^{19}-\frac{2}{3375}a^{15}-\frac{2}{2625}a^{13}+\frac{2}{675}a^{11}-\frac{1}{175}a^{9}+\frac{31}{945}a^{7}+\frac{8}{105}a^{5}-\frac{31}{189}a^{3}-\frac{1}{7}a$, $\frac{1}{611296875}a^{24}+\frac{13}{40753125}a^{22}-\frac{11}{17465625}a^{20}-\frac{64}{8150625}a^{18}-\frac{208}{24451875}a^{16}+\frac{2101}{1630125}a^{14}-\frac{1982}{4890375}a^{12}-\frac{83}{65205}a^{10}+\frac{683}{139725}a^{8}-\frac{610}{13041}a^{6}+\frac{4723}{195615}a^{4}-\frac{853}{13041}a^{2}-\frac{1763}{5589}$, $\frac{1}{611296875}a^{25}+\frac{13}{40753125}a^{23}-\frac{11}{17465625}a^{21}-\frac{64}{8150625}a^{19}-\frac{208}{24451875}a^{17}+\frac{2101}{1630125}a^{15}-\frac{1982}{4890375}a^{13}-\frac{83}{65205}a^{11}+\frac{683}{139725}a^{9}-\frac{610}{13041}a^{7}+\frac{4723}{195615}a^{5}-\frac{853}{13041}a^{3}-\frac{1763}{5589}a$, $\frac{1}{97\!\cdots\!25}a^{26}+\frac{74\!\cdots\!52}{97\!\cdots\!25}a^{24}+\frac{35\!\cdots\!26}{78\!\cdots\!25}a^{22}+\frac{11\!\cdots\!19}{19\!\cdots\!25}a^{20}-\frac{79\!\cdots\!51}{39\!\cdots\!25}a^{18}+\frac{38\!\cdots\!74}{30\!\cdots\!25}a^{16}+\frac{96\!\cdots\!22}{78\!\cdots\!25}a^{14}-\frac{37\!\cdots\!51}{78\!\cdots\!25}a^{12}+\frac{53\!\cdots\!97}{15\!\cdots\!25}a^{10}-\frac{37\!\cdots\!73}{12\!\cdots\!25}a^{8}+\frac{93\!\cdots\!85}{62\!\cdots\!21}a^{6}-\frac{44\!\cdots\!59}{44\!\cdots\!15}a^{4}-\frac{22\!\cdots\!01}{62\!\cdots\!21}a^{2}-\frac{59\!\cdots\!01}{62\!\cdots\!21}$, $\frac{1}{97\!\cdots\!25}a^{27}+\frac{74\!\cdots\!52}{97\!\cdots\!25}a^{25}+\frac{35\!\cdots\!26}{78\!\cdots\!25}a^{23}+\frac{11\!\cdots\!19}{19\!\cdots\!25}a^{21}-\frac{79\!\cdots\!51}{39\!\cdots\!25}a^{19}+\frac{38\!\cdots\!74}{30\!\cdots\!25}a^{17}+\frac{96\!\cdots\!22}{78\!\cdots\!25}a^{15}-\frac{37\!\cdots\!51}{78\!\cdots\!25}a^{13}+\frac{53\!\cdots\!97}{15\!\cdots\!25}a^{11}-\frac{37\!\cdots\!73}{12\!\cdots\!25}a^{9}+\frac{93\!\cdots\!85}{62\!\cdots\!21}a^{7}-\frac{44\!\cdots\!59}{44\!\cdots\!15}a^{5}-\frac{22\!\cdots\!01}{62\!\cdots\!21}a^{3}-\frac{59\!\cdots\!01}{62\!\cdots\!21}a$
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
\(\Q(\sqrt{5}) \), 4.2.302000.2, 7.1.3442951.1, 14.2.926086842843828125.1 |
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 | ${\href{/padicField/3.2.0.1}{2} }^{14}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{14}$ | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{14}$ | $28$ | ${\href{/padicField/19.7.0.1}{7} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{14}$ | ${\href{/padicField/29.7.0.1}{7} }^{4}$ | ${\href{/padicField/31.7.0.1}{7} }^{4}$ | $28$ | ${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $28$ | $28$ | ${\href{/padicField/53.2.0.1}{2} }^{14}$ | ${\href{/padicField/59.14.0.1}{14} }^{2}$ |
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 $28$ | $2$ | $14$ | $28$ | |||
\(5\) | Deg $28$ | $4$ | $7$ | $21$ | |||
\(151\) | $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | 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 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.755.2t1.a.a | $1$ | $ 5 \cdot 151 $ | \(\Q(\sqrt{-755}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.151.2t1.a.a | $1$ | $ 151 $ | \(\Q(\sqrt{-151}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 2.60400.4t3.c.a | $2$ | $ 2^{4} \cdot 5^{2} \cdot 151 $ | 4.0.45602000.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 2.151.7t2.a.a | $2$ | $ 151 $ | 7.1.3442951.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.3775.14t3.a.c | $2$ | $ 5^{2} \cdot 151 $ | 14.0.139839113269418046875.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.3775.14t3.a.a | $2$ | $ 5^{2} \cdot 151 $ | 14.0.139839113269418046875.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.151.7t2.a.c | $2$ | $ 151 $ | 7.1.3442951.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.151.7t2.a.b | $2$ | $ 151 $ | 7.1.3442951.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.3775.14t3.a.b | $2$ | $ 5^{2} \cdot 151 $ | 14.0.139839113269418046875.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.60400.28t10.a.a | $2$ | $ 2^{4} \cdot 5^{2} \cdot 151 $ | 28.2.2715878171109088684095776889728000000000000000000000.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
* | 2.60400.28t10.a.e | $2$ | $ 2^{4} \cdot 5^{2} \cdot 151 $ | 28.2.2715878171109088684095776889728000000000000000000000.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
* | 2.60400.28t10.a.b | $2$ | $ 2^{4} \cdot 5^{2} \cdot 151 $ | 28.2.2715878171109088684095776889728000000000000000000000.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
* | 2.60400.28t10.a.f | $2$ | $ 2^{4} \cdot 5^{2} \cdot 151 $ | 28.2.2715878171109088684095776889728000000000000000000000.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
* | 2.60400.28t10.a.c | $2$ | $ 2^{4} \cdot 5^{2} \cdot 151 $ | 28.2.2715878171109088684095776889728000000000000000000000.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
* | 2.60400.28t10.a.d | $2$ | $ 2^{4} \cdot 5^{2} \cdot 151 $ | 28.2.2715878171109088684095776889728000000000000000000000.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |