Normalized defining polynomial
\( x^{8} - x^{7} - 13x^{6} + 17x^{5} + 40x^{4} - 62x^{3} - 13x^{2} + 31x + 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(6891328125\) \(\medspace = 3^{6}\cdot 5^{7}\cdot 11^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(16.97\) | 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))
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Galois root discriminant: | $3^{3/4}5^{7/8}11^{1/2}\approx 30.912635812282684$ | ||
Ramified primes: | \(3\), \(5\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{421}a^{7}-\frac{206}{421}a^{6}+\frac{117}{421}a^{5}+\frac{29}{421}a^{4}-\frac{11}{421}a^{3}+\frac{88}{421}a^{2}+\frac{50}{421}a-\frac{115}{421}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{49}{421}a^{7}+\frac{10}{421}a^{6}-\frac{582}{421}a^{5}+\frac{158}{421}a^{4}+\frac{1566}{421}a^{3}-\frac{1161}{421}a^{2}-\frac{76}{421}a+\frac{680}{421}$, $\frac{84}{421}a^{7}-\frac{43}{421}a^{6}-\frac{1118}{421}a^{5}+\frac{752}{421}a^{4}+\frac{3707}{421}a^{3}-\frac{2291}{421}a^{2}-\frac{2115}{421}a+\frac{444}{421}$, $\frac{198}{421}a^{7}+\frac{49}{421}a^{6}-\frac{2515}{421}a^{5}+\frac{269}{421}a^{4}+\frac{8347}{421}a^{3}-\frac{2363}{421}a^{2}-\frac{5677}{421}a-\frac{36}{421}$, $\frac{223}{421}a^{7}-\frac{49}{421}a^{6}-\frac{2958}{421}a^{5}+\frac{1415}{421}a^{4}+\frac{10177}{421}a^{3}-\frac{5215}{421}a^{2}-\frac{6953}{421}a+\frac{36}{421}$, $\frac{192}{421}a^{7}+\frac{22}{421}a^{6}-\frac{2375}{421}a^{5}+\frac{516}{421}a^{4}+\frac{7150}{421}a^{3}-\frac{2470}{421}a^{2}-\frac{3030}{421}a-\frac{609}{421}$, $\frac{23}{421}a^{7}-\frac{107}{421}a^{6}-\frac{256}{421}a^{5}+\frac{1509}{421}a^{4}+\frac{168}{421}a^{3}-\frac{5133}{421}a^{2}+\frac{1992}{421}a+\frac{2828}{421}$, $\frac{114}{421}a^{7}+\frac{92}{421}a^{6}-\frac{1397}{421}a^{5}-\frac{483}{421}a^{4}+\frac{4640}{421}a^{3}-\frac{72}{421}a^{2}-\frac{3983}{421}a+\frac{362}{421}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 161.626969366 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 161.626969366 \cdot 1}{2\cdot\sqrt{6891328125}}\cr\approx \mathstrut & 0.249213942001 \end{aligned}\]
Galois group
$\OD_{16}$ (as 8T7):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_8:C_2$ |
Character table for $C_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 16 |
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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | R | ${\href{/padicField/7.8.0.1}{8} }$ | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.3 | $x^{8} - 6 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
\(5\) | 5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
\(11\) | 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
11.4.2.2 | $x^{4} - 77 x^{2} + 242$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.33.2t1.a.a | $1$ | $ 3 \cdot 11 $ | \(\Q(\sqrt{33}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.165.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 11 $ | \(\Q(\sqrt{165}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
1.55.4t1.a.a | $1$ | $ 5 \cdot 11 $ | 4.4.15125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.55.4t1.a.b | $1$ | $ 5 \cdot 11 $ | 4.4.15125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
* | 2.2475.8t7.a.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 11 $ | 8.8.6891328125.1 | $C_8:C_2$ (as 8T7) | $0$ | $2$ |
* | 2.2475.8t7.a.b | $2$ | $ 3^{2} \cdot 5^{2} \cdot 11 $ | 8.8.6891328125.1 | $C_8:C_2$ (as 8T7) | $0$ | $2$ |