Normalized defining polynomial
\( x^{8} - 2x^{7} - 38x^{6} + 2x^{5} + 458x^{4} + 562x^{3} - 1438x^{2} - 3402x - 1899 \)
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: | \(333621760000\) \(\medspace = 2^{12}\cdot 5^{4}\cdot 19^{4}\) | 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: | \(27.57\) | 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: | $2^{3/2}5^{1/2}19^{1/2}\approx 27.568097504180443$ | ||
Ramified primes: | \(2\), \(5\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{1}{3}a^{2}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{114}a^{6}-\frac{3}{38}a^{5}+\frac{4}{57}a^{4}+\frac{2}{57}a^{3}-\frac{29}{114}a^{2}+\frac{13}{114}a-\frac{8}{19}$, $\frac{1}{114}a^{7}+\frac{1}{38}a^{5}-\frac{1}{6}a^{4}+\frac{7}{114}a^{3}+\frac{28}{57}a^{2}-\frac{7}{114}a-\frac{11}{38}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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: | $\frac{1}{38}a^{7}-\frac{13}{114}a^{6}-\frac{83}{114}a^{5}+\frac{100}{57}a^{4}+\frac{881}{114}a^{3}-\frac{135}{38}a^{2}-\frac{167}{6}a-\frac{397}{19}$, $\frac{7}{38}a^{7}-\frac{13}{19}a^{6}-\frac{110}{19}a^{5}+\frac{381}{38}a^{4}+\frac{2529}{38}a^{3}-\frac{111}{19}a^{2}-\frac{4725}{19}a-\frac{8369}{38}$, $\frac{1}{114}a^{7}+\frac{1}{19}a^{6}-\frac{89}{114}a^{5}-\frac{41}{38}a^{4}+\frac{1285}{114}a^{3}+\frac{240}{19}a^{2}-\frac{5021}{114}a-\frac{2083}{38}$, $\frac{9}{38}a^{7}-\frac{55}{57}a^{6}-\frac{791}{114}a^{5}+\frac{549}{38}a^{4}+\frac{8869}{114}a^{3}-\frac{1259}{57}a^{2}-\frac{10977}{38}a-\frac{461}{2}$, $\frac{17}{114}a^{7}-\frac{21}{38}a^{6}-\frac{541}{114}a^{5}+\frac{163}{19}a^{4}+\frac{323}{6}a^{3}-\frac{429}{38}a^{2}-\frac{22465}{114}a-\frac{2924}{19}$, $\frac{4}{19}a^{7}-\frac{15}{19}a^{6}-\frac{376}{57}a^{5}+\frac{1351}{114}a^{4}+\frac{1431}{19}a^{3}-\frac{664}{57}a^{2}-\frac{15926}{57}a-\frac{8875}{38}$, $\frac{13}{114}a^{7}-\frac{29}{38}a^{6}-\frac{34}{19}a^{5}+\frac{811}{57}a^{4}+\frac{1111}{114}a^{3}-\frac{8947}{114}a^{2}-\frac{1238}{57}a+\frac{2135}{19}$ | 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: | \( 1758.1037263 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 1758.1037263 \cdot 1}{2\cdot\sqrt{333621760000}}\cr\approx \mathstrut & 0.38960747397 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{95}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{19})\), 4.4.115520.1 x2, 4.4.7600.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.4.115520.1, 4.4.7600.1 |
Minimal sibling: | 4.4.7600.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.14 | $x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$ | $4$ | $2$ | $12$ | $D_4$ | $[2, 2]^{2}$ |
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(19\) | 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.76.2t1.a.a | $1$ | $ 2^{2} \cdot 19 $ | \(\Q(\sqrt{19}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.380.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 19 $ | \(\Q(\sqrt{95}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.1520.4t3.e.a | $2$ | $ 2^{4} \cdot 5 \cdot 19 $ | 8.8.333621760000.1 | $D_4$ (as 8T4) | $1$ | $2$ |