Normalized defining polynomial
\( x^{8} - 4x^{7} + 13x^{6} - 25x^{5} + 40x^{4} - 43x^{3} + 10x^{2} + 8x + 1 \)
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: | \(468838125\) \(\medspace = 3^{7}\cdot 5^{4}\cdot 7^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.13\) | 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: | $3^{7/8}5^{1/2}7^{3/4}\approx 25.16457673743996$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{21}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{1}{7}a^{2}-\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{49}a^{7}+\frac{13}{49}a^{5}-\frac{22}{49}a^{4}+\frac{1}{49}a^{3}+\frac{10}{49}a^{2}+\frac{1}{49}a+\frac{12}{49}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{3}{7} a^{6} + \frac{9}{7} a^{5} - \frac{31}{7} a^{4} + \frac{47}{7} a^{3} - \frac{81}{7} a^{2} + \frac{59}{7} a + \frac{16}{7} \) (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: | $\frac{29}{49}a^{7}-\frac{13}{7}a^{6}+\frac{307}{49}a^{5}-\frac{484}{49}a^{4}+\frac{834}{49}a^{3}-\frac{648}{49}a^{2}-\frac{27}{49}a+\frac{33}{49}$, $\frac{3}{7}a^{7}-\frac{12}{7}a^{6}+\frac{40}{7}a^{5}-\frac{78}{7}a^{4}+\frac{128}{7}a^{3}-20a^{2}+\frac{50}{7}a+\frac{9}{7}$, $a$ | 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: | \( 22.3835917465 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 22.3835917465 \cdot 2}{6\cdot\sqrt{468838125}}\cr\approx \mathstrut & 0.537052242857 \end{aligned}\]
Galois group
A solvable group of order 64 |
The 16 conjugacy class representatives for $(C_4^2 : C_2):C_2$ |
Character table for $(C_4^2 : C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.4725.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Minimal sibling: | 8.0.18753525.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }$ | R | R | R | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.7.1 | $x^{8} + 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
\(5\) | 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.4.3.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.105.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 7 $ | \(\Q(\sqrt{105}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.735.4t3.f.a | $2$ | $ 3 \cdot 5 \cdot 7^{2}$ | 4.0.25725.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.2205.4t3.a.a | $2$ | $ 3^{2} \cdot 5 \cdot 7^{2}$ | 4.0.231525.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
2.2205.4t3.b.a | $2$ | $ 3^{2} \cdot 5 \cdot 7^{2}$ | 4.4.231525.1 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
2.735.4t3.e.a | $2$ | $ 3 \cdot 5 \cdot 7^{2}$ | 4.0.25725.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.63.4t3.a.a | $2$ | $ 3^{2} \cdot 7 $ | 4.0.189.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.1575.4t3.a.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 4.0.4725.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
4.4862025.8t26.d.a | $4$ | $ 3^{4} \cdot 5^{2} \cdot 7^{4}$ | 8.0.468838125.1 | $(C_4^2 : C_2):C_2$ (as 8T26) | $1$ | $0$ | |
* | 4.99225.8t26.d.a | $4$ | $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ | 8.0.468838125.1 | $(C_4^2 : C_2):C_2$ (as 8T26) | $1$ | $0$ |