Normalized defining polynomial
\( x^{8} - x^{7} + 5x^{6} + x^{5} - 23x^{4} + 9x^{3} + 27x^{2} - 9x + 14 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(5482378736704\) \(\medspace = 2^{6}\cdot 541^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.12\) | 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: | $2^{6/7}541^{2/3}\approx 120.26997498474019$ | ||
Ramified primes: | \(2\), \(541\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{148}a^{7}+\frac{9}{74}a^{6}-\frac{23}{148}a^{5}+\frac{2}{37}a^{4}-\frac{19}{148}a^{3}-\frac{14}{37}a^{2}+\frac{73}{148}a+\frac{23}{74}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $3$ | 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: | $\frac{58}{37}a^{7}+\frac{8}{37}a^{6}+\frac{220}{37}a^{5}+\frac{743}{74}a^{4}-\frac{2537}{74}a^{3}-\frac{2167}{74}a^{2}+\frac{4435}{74}a+\frac{1780}{37}$, $\frac{8255}{148}a^{7}-\frac{4885}{74}a^{6}+\frac{39239}{148}a^{5}+\frac{970}{37}a^{4}-\frac{207461}{148}a^{3}+\frac{25770}{37}a^{2}+\frac{295959}{148}a-\frac{67729}{74}$, $\frac{4205}{148}a^{7}+\frac{14609}{74}a^{6}-\frac{11393}{148}a^{5}+\frac{82569}{74}a^{4}-\frac{62653}{148}a^{3}-\frac{369969}{74}a^{2}+\frac{334345}{148}a+\frac{502605}{74}$ (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: | \( 7746.14226788 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 7746.14226788 \cdot 1}{2\cdot\sqrt{5482378736704}}\cr\approx \mathstrut & 2.57804461969 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 168 |
The 8 conjugacy class representatives for $C_2^3:(C_7: C_3)$ |
Character table for $C_2^3:(C_7: C_3)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 14 sibling: | deg 14 |
Degree 24 sibling: | deg 24 |
Degree 28 sibling: | deg 28 |
Degree 42 sibling: | 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 | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(541\) | $\Q_{541}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{541}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $3$ | $3$ | $1$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.541.3t1.a.a | $1$ | $ 541 $ | 3.3.292681.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.541.3t1.a.b | $1$ | $ 541 $ | 3.3.292681.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
3.2341448.7t3.a.a | $3$ | $ 2^{3} \cdot 541^{2}$ | 7.7.5482378736704.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
3.2341448.7t3.a.b | $3$ | $ 2^{3} \cdot 541^{2}$ | 7.7.5482378736704.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
* | 7.548...704.8t36.c.a | $7$ | $ 2^{6} \cdot 541^{4}$ | 8.0.5482378736704.2 | $C_2^3:(C_7: C_3)$ (as 8T36) | $1$ | $-1$ |
7.296...864.24t283.c.a | $7$ | $ 2^{6} \cdot 541^{5}$ | 8.0.5482378736704.2 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ | |
7.296...864.24t283.c.b | $7$ | $ 2^{6} \cdot 541^{5}$ | 8.0.5482378736704.2 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ |