Normalized defining polynomial
\( x^{8} - 4x^{7} + 9x^{6} - 5x^{5} - 12x^{3} + 5x^{2} + 3x - 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(12834170944\) \(\medspace = 2^{6}\cdot 7^{4}\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.35\) | 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: | $2^{3/2}7^{2/3}17^{1/2}\approx 42.67447112735335$ | ||
Ramified primes: | \(2\), \(7\), \(17\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{74}a^{7}-\frac{33}{74}a^{6}+\frac{2}{37}a^{5}+\frac{27}{74}a^{4}+\frac{31}{74}a^{3}-\frac{23}{74}a^{2}+\frac{3}{37}a-\frac{23}{74}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $a^{7}-4a^{6}+9a^{5}-5a^{4}-12a^{2}+5a+3$, $\frac{127}{37}a^{7}-\frac{454}{37}a^{6}+\frac{952}{37}a^{5}-\frac{234}{37}a^{4}-\frac{96}{37}a^{3}-\frac{1552}{37}a^{2}-\frac{15}{37}a+\frac{335}{37}$, $\frac{81}{37}a^{7}-\frac{305}{37}a^{6}+\frac{657}{37}a^{5}-\frac{255}{37}a^{4}-\frac{42}{37}a^{3}-\frac{1012}{37}a^{2}+\frac{153}{37}a+\frac{246}{37}$, $\frac{18}{37}a^{7}-\frac{76}{37}a^{6}+\frac{183}{37}a^{5}-\frac{143}{37}a^{4}+\frac{40}{37}a^{3}-\frac{155}{37}a^{2}-\frac{3}{37}a+\frac{67}{37}$, $\frac{97}{37}a^{7}-\frac{352}{37}a^{6}+\frac{758}{37}a^{5}-\frac{267}{37}a^{4}+\frac{47}{37}a^{3}-\frac{1232}{37}a^{2}+\frac{64}{37}a+\frac{100}{37}$ | 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: | \( 342.972806567 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 342.972806567 \cdot 1}{2\cdot\sqrt{12834170944}}\cr\approx \mathstrut & 0.956148837271 \end{aligned}\]
Galois group
$C_2^3:A_4$ (as 8T33):
A solvable group of order 96 |
The 10 conjugacy class representatives for $C_2^4:C_6$ |
Character table for $C_2^4:C_6$ |
Intermediate fields
\(\Q(\sqrt{17}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $[\ ]$ |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
2.4.6.7 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(17\) | 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.6.3.1 | $x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.119.6t1.b.a | $1$ | $ 7 \cdot 17 $ | 6.6.11796113.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.119.6t1.b.b | $1$ | $ 7 \cdot 17 $ | 6.6.11796113.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
3.3136.4t4.a.a | $3$ | $ 2^{6} \cdot 7^{2}$ | 4.0.3136.1 | $A_4$ (as 4T4) | $1$ | $-1$ | |
3.15407168.6t6.a.a | $3$ | $ 2^{6} \cdot 7^{2} \cdot 17^{3}$ | 6.2.754951232.2 | $A_4\times C_2$ (as 6T6) | $1$ | $-1$ | |
6.48316878848.8t33.a.a | $6$ | $ 2^{12} \cdot 7^{4} \cdot 17^{3}$ | 8.4.12834170944.2 | $C_2^4:C_6$ (as 8T33) | $1$ | $-2$ | |
* | 6.754951232.8t33.a.a | $6$ | $ 2^{6} \cdot 7^{4} \cdot 17^{3}$ | 8.4.12834170944.2 | $C_2^4:C_6$ (as 8T33) | $1$ | $2$ |