Normalized defining polynomial
\( x^{8} - 2x^{7} + 14x^{6} - 14x^{5} - 126x^{4} + 630x^{3} + 658x^{2} - 5130x + 10267 \)
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: | \(226775649501184\) \(\medspace = 2^{14}\cdot 7^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(62.29\) | 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: | $2^{7/4}7^{12/7}\approx 94.52397781331281$ | ||
Ramified primes: | \(2\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{26847444391}a^{7}-\frac{3241496060}{26847444391}a^{6}+\frac{659736697}{1413023389}a^{5}-\frac{8539178089}{26847444391}a^{4}+\frac{7541133610}{26847444391}a^{3}-\frac{2893722651}{26847444391}a^{2}-\frac{167708978}{26847444391}a-\frac{3957389386}{26847444391}$
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: | $\frac{14203445540}{26847444391}a^{7}+\frac{907308787}{26847444391}a^{6}+\frac{8632871460}{1413023389}a^{5}+\frac{34320240974}{26847444391}a^{4}-\frac{2442936625098}{26847444391}a^{3}+\frac{1804053145017}{26847444391}a^{2}+\frac{13337074712054}{26847444391}a-\frac{48771907844403}{26847444391}$, $\frac{565465835}{26847444391}a^{7}-\frac{632255401}{26847444391}a^{6}+\frac{451790503}{1413023389}a^{5}+\frac{2785393762}{26847444391}a^{4}-\frac{143706612855}{26847444391}a^{3}+\frac{880142975629}{26847444391}a^{2}-\frac{1838679101271}{26847444391}a+\frac{2106227928585}{26847444391}$, $\frac{2417306133420}{26847444391}a^{7}+\frac{9600629328573}{26847444391}a^{6}+\frac{2900515640420}{1413023389}a^{5}+\frac{191420615374470}{26847444391}a^{4}+\frac{129753125076564}{26847444391}a^{3}+\frac{216191707231287}{26847444391}a^{2}+\frac{32\!\cdots\!88}{26847444391}a+\frac{32\!\cdots\!49}{26847444391}$ (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: | \( 24239.5513308 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 24239.5513308 \cdot 1}{2\cdot\sqrt{226775649501184}}\cr\approx \mathstrut & 1.25434165516 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 56 |
The 8 conjugacy class representatives for $C_2^3:C_7$ |
Character table for $C_2^3:C_7$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 14 sibling: | deg 14 |
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 | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.14.6 | $x^{8} + 2 x^{7} + 2 x^{4} + 6$ | $8$ | $1$ | $14$ | $C_2^3:C_7$ | $[2, 2, 2]^{7}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.49.7t1.a.a | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
1.49.7t1.a.b | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
1.49.7t1.a.c | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
1.49.7t1.a.d | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
1.49.7t1.a.e | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
1.49.7t1.a.f | $1$ | $ 7^{2}$ | 7.7.13841287201.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
* | 7.226...184.8t25.b.a | $7$ | $ 2^{14} \cdot 7^{12}$ | 8.0.226775649501184.4 | $C_2^3:C_7$ (as 8T25) | $1$ | $-1$ |