Normalized defining polynomial
\( x^{28} - 3x^{14} + 3 \)
Invariants
Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(941583580117685398903952998389423397952028672\) \(\medspace = 2^{28}\cdot 3^{27}\cdot 7^{28}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(40.38\) | 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: | $2^{3/2}3^{27/28}7^{47/42}\approx 71.99977726879091$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$
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)
| |
Torsion generator: | \( -a^{14} + 2 \) (order $6$) | 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: | $a^{14}-a^{2}-1$, $a^{16}-a^{14}-a^{2}+1$, $a^{15}+a^{14}-2a-2$, $a^{7}+1$, $a^{22}+a^{16}+a^{14}+a^{10}-a^{8}+a^{4}-a^{2}-1$, $a^{26}-a^{22}-a^{20}+a^{14}-a^{12}+a^{10}+2a^{8}+a^{6}-a^{4}-a^{2}-2$, $a^{26}-a^{22}+a^{14}-a^{12}+2a^{8}-a^{4}-a^{2}-1$, $a^{26}-a^{22}-a^{16}+a^{14}-a^{12}+2a^{8}-a^{4}+a^{2}-1$, $a^{26}+a^{25}+a^{23}+a^{22}+a^{20}+a^{19}+a^{17}+a^{16}+a^{14}-2a^{12}-a^{11}-2a^{9}-a^{8}-2a^{6}-a^{5}-2a^{3}-a^{2}-2$, $a^{26}+2a^{25}-a^{23}-2a^{22}-a^{21}+a^{20}+2a^{18}-a^{16}-a^{15}-2a^{14}+a^{13}-a^{12}-3a^{11}+3a^{8}+2a^{7}+a^{5}-4a^{4}-2a^{3}+3a+4$, $a^{27}-a^{25}-a^{24}+2a^{23}-a^{21}-2a^{20}+a^{19}+2a^{18}-a^{17}-a^{15}+a^{14}-2a^{13}+a^{12}+a^{11}-4a^{9}+a^{8}+3a^{7}+a^{6}-2a^{5}-3a^{4}+3a^{3}-2$, $a^{27}-a^{25}-a^{24}+2a^{23}+3a^{22}+a^{21}-3a^{20}-2a^{19}+2a^{17}-a^{16}-a^{15}-a^{14}-a^{13}+a^{12}+2a^{11}-5a^{9}-4a^{8}+a^{7}+6a^{6}+a^{5}-2a^{4}-3a^{3}+3a^{2}+2a+1$, $a^{27}-a^{26}-3a^{25}-a^{24}+a^{23}+3a^{22}-a^{20}-a^{19}-a^{17}-a^{16}+a^{15}+4a^{14}+a^{12}+4a^{11}+a^{10}-2a^{9}-6a^{8}+3a^{6}+4a^{5}-a^{3}-a^{2}-a-5$ (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)]
| |
Regulator: | \( 647468634170.5619 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 647468634170.5619 \cdot 1}{6\cdot\sqrt{941583580117685398903952998389423397952028672}}\cr\approx \mathstrut & 0.525601024181358 \end{aligned}\] (assuming GRH)
Galois group
$D_4\times F_7$ (as 28T41):
A solvable group of order 336 |
The 35 conjugacy class representatives for $D_4\times F_7$ |
Character table for $D_4\times F_7$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.432.1, 7.1.600362847.1, 14.0.1081306644173836227.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 28 siblings: | data not computed |
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 | R | ${\href{/padicField/5.12.0.1}{12} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{14}$ | ${\href{/padicField/17.12.0.1}{12} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.6.0.1}{6} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $28$ | ${\href{/padicField/31.6.0.1}{6} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{7}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
2.12.12.18 | $x^{12} - 10 x^{11} + 72 x^{10} - 332 x^{9} + 1316 x^{8} - 4160 x^{7} + 12128 x^{6} - 27904 x^{5} + 53744 x^{4} - 69600 x^{3} + 71680 x^{2} - 41536 x + 38848$ | $2$ | $6$ | $12$ | $D_4 \times C_3$ | $[2, 2]^{6}$ | |
2.12.12.18 | $x^{12} - 10 x^{11} + 72 x^{10} - 332 x^{9} + 1316 x^{8} - 4160 x^{7} + 12128 x^{6} - 27904 x^{5} + 53744 x^{4} - 69600 x^{3} + 71680 x^{2} - 41536 x + 38848$ | $2$ | $6$ | $12$ | $D_4 \times C_3$ | $[2, 2]^{6}$ | |
\(3\) | Deg $28$ | $28$ | $1$ | $27$ | |||
\(7\) | 7.7.7.5 | $x^{7} + 7 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ |
7.7.7.5 | $x^{7} + 7 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ | |
7.14.14.21 | $x^{14} - 14 x^{9} + 14 x^{8} + 14 x^{7} - 1127 x^{4} - 98 x^{3} - 49 x^{2} + 98 x + 49$ | $7$ | $2$ | $14$ | $F_7 \times C_2$ | $[7/6]_{6}^{2}$ |