Normalized defining polynomial
\( x^{8} - 2x^{7} - 17x^{6} + 16x^{5} + 100x^{4} + x^{3} - 197x^{2} - 152x - 29 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(20560078125\) \(\medspace = 3^{6}\cdot 5^{7}\cdot 19^{2}\) | 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: | \(19.46\) | 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: | $3^{3/4}5^{7/8}19^{1/2}\approx 40.62716288440264$ | ||
Ramified primes: | \(3\), \(5\), \(19\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{87}a^{7}+\frac{11}{87}a^{6}+\frac{10}{87}a^{5}+\frac{1}{87}a^{4}-\frac{32}{87}a^{3}+\frac{20}{87}a^{2}+\frac{34}{87}a$
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)
| |
Fundamental units: | $\frac{8}{87}a^{7}-\frac{28}{87}a^{6}-\frac{94}{87}a^{5}+\frac{269}{87}a^{4}+\frac{137}{29}a^{3}-\frac{623}{87}a^{2}-\frac{772}{87}a-\frac{4}{3}$, $\frac{2}{29}a^{7}-\frac{7}{29}a^{6}-\frac{56}{87}a^{5}+\frac{60}{29}a^{4}+\frac{98}{87}a^{3}-\frac{373}{87}a^{2}+\frac{39}{29}a+\frac{5}{3}$, $\frac{7}{87}a^{7}-\frac{10}{87}a^{6}-\frac{133}{87}a^{5}+\frac{94}{87}a^{4}+\frac{254}{29}a^{3}-\frac{92}{87}a^{2}-\frac{1328}{87}a-\frac{22}{3}$, $\frac{7}{29}a^{7}-\frac{59}{87}a^{6}-\frac{104}{29}a^{5}+\frac{572}{87}a^{4}+\frac{1706}{87}a^{3}-\frac{353}{29}a^{2}-\frac{3491}{87}a-18$, $\frac{4}{29}a^{7}-\frac{14}{29}a^{6}-\frac{47}{29}a^{5}+\frac{389}{87}a^{4}+\frac{631}{87}a^{3}-\frac{268}{29}a^{2}-\frac{1187}{87}a-\frac{11}{3}$, $\frac{7}{87}a^{7}-\frac{13}{29}a^{6}-\frac{46}{87}a^{5}+\frac{128}{29}a^{4}+\frac{182}{87}a^{3}-\frac{875}{87}a^{2}-\frac{835}{87}a-\frac{7}{3}$, $\frac{14}{29}a^{7}-\frac{118}{87}a^{6}-\frac{595}{87}a^{5}+\frac{1115}{87}a^{4}+\frac{2977}{87}a^{3}-\frac{2234}{87}a^{2}-\frac{5387}{87}a-16$ | 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: | \( 299.59656431 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 299.59656431 \cdot 1}{2\cdot\sqrt{20560078125}}\cr\approx \mathstrut & 0.26744496001 \end{aligned}\]
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 | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | R | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
\(19\) | 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
19.4.2.2 | $x^{4} - 2888 x^{3} - 767106 x^{2} - 76532 x + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.285.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 19 $ | \(\Q(\sqrt{285}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.57.2t1.a.a | $1$ | $ 3 \cdot 19 $ | \(\Q(\sqrt{57}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.95.4t1.a.a | $1$ | $ 5 \cdot 19 $ | 4.4.45125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
* | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
1.95.4t1.a.b | $1$ | $ 5 \cdot 19 $ | 4.4.45125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 2.4275.8t7.a.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 19 $ | 8.8.20560078125.1 | $C_8:C_2$ (as 8T7) | $0$ | $2$ |
* | 2.4275.8t7.a.b | $2$ | $ 3^{2} \cdot 5^{2} \cdot 19 $ | 8.8.20560078125.1 | $C_8:C_2$ (as 8T7) | $0$ | $2$ |