Normalized defining polynomial
\( x^{8} - 4x^{7} - 4x^{6} + 20x^{5} - 29x^{4} + 58x^{3} + 152x^{2} - 74x + 24 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(10504456959364\) \(\medspace = 2^{2}\cdot 19^{4}\cdot 67^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(42.43\) | 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: | $2^{2/3}19^{1/2}67^{1/2}\approx 56.637081431093776$ | ||
Ramified primes: | \(2\), \(19\), \(67\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{9816}a^{7}+\frac{631}{9816}a^{6}-\frac{1775}{9816}a^{5}+\frac{1735}{9816}a^{4}+\frac{96}{409}a^{3}+\frac{257}{4908}a^{2}+\frac{1307}{4908}a+\frac{38}{409}$
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)
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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: | $\frac{29}{4908}a^{7}-\frac{53}{2454}a^{6}+\frac{59}{4908}a^{5}+\frac{2}{1227}a^{4}-\frac{158}{409}a^{3}+\frac{2545}{2454}a^{2}-\frac{1294}{1227}a+\frac{159}{409}$, $\frac{125}{9816}a^{7}+\frac{347}{9816}a^{6}-\frac{1015}{9816}a^{5}-\frac{3985}{9816}a^{4}-\frac{270}{409}a^{3}-\frac{7139}{4908}a^{2}-\frac{8405}{4908}a-\frac{158}{409}$, $\frac{785}{4908}a^{7}-\frac{2827}{4908}a^{6}-\frac{4411}{4908}a^{5}+\frac{14729}{4908}a^{4}-\frac{1428}{409}a^{3}+\frac{17695}{2454}a^{2}+\frac{68935}{2454}a-\frac{463}{409}$, $\frac{235}{3272}a^{7}-\frac{1409}{3272}a^{6}+\frac{1691}{3272}a^{5}+\frac{2815}{3272}a^{4}-\frac{1850}{409}a^{3}+\frac{17859}{1636}a^{2}-\frac{7785}{1636}a+\frac{614}{409}$, $\frac{692259}{3272}a^{7}-\frac{3791273}{3272}a^{6}-\frac{2414125}{3272}a^{5}+\frac{33630399}{3272}a^{4}-\frac{1854954}{409}a^{3}-\frac{31067169}{1636}a^{2}+\frac{15173975}{1636}a-\frac{1131136}{409}$ | 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: | \( 22253.0500822 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 22253.0500822 \cdot 1}{2\cdot\sqrt{10504456959364}}\cr\approx \mathstrut & 2.16846524589 \end{aligned}\]
Galois group
$\PGOPlus(4,3)$ (as 8T45):
A solvable group of order 576 |
The 16 conjugacy class representatives for $(A_4\wr C_2):C_2$ |
Character table for $(A_4\wr C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{1273}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 36 siblings: | 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 | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | R | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $[\ ]$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(19\) | 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.6.3.2 | $x^{6} + 65 x^{4} + 34 x^{3} + 1099 x^{2} - 1802 x + 4564$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(67\) | 67.8.4.1 | $x^{8} + 284 x^{6} + 108 x^{5} + 28074 x^{4} - 13608 x^{3} + 1141144 x^{2} - 1396332 x + 15837397$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.67.2t1.a.a | $1$ | $ 67 $ | \(\Q(\sqrt{-67}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.1273.2t1.a.a | $1$ | $ 19 \cdot 67 $ | \(\Q(\sqrt{1273}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.268.3t2.a.a | $2$ | $ 2^{2} \cdot 67 $ | 3.1.268.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.341164.6t3.a.a | $2$ | $ 2^{2} \cdot 19 \cdot 67^{2}$ | 6.2.33006934672.2 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.76.3t2.a.a | $2$ | $ 2^{2} \cdot 19 $ | 3.1.76.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.96748.6t3.a.a | $2$ | $ 2^{2} \cdot 19^{2} \cdot 67 $ | 6.2.33006934672.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
4.6482116.6t9.a.a | $4$ | $ 2^{2} \cdot 19^{2} \cdot 67^{2}$ | 6.2.8251733668.1 | $S_3^2$ (as 6T9) | $1$ | $0$ | |
* | 6.8251733668.8t45.a.a | $6$ | $ 2^{2} \cdot 19^{3} \cdot 67^{3}$ | 8.4.10504456959364.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $2$ |
6.8251733668.12t161.a.a | $6$ | $ 2^{2} \cdot 19^{3} \cdot 67^{3}$ | 8.4.10504456959364.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $-2$ | |
9.397...944.18t185.a.a | $9$ | $ 2^{6} \cdot 19^{3} \cdot 67^{6}$ | 8.4.10504456959364.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $-1$ | |
9.132027738688.12t165.a.a | $9$ | $ 2^{6} \cdot 19^{3} \cdot 67^{3}$ | 8.4.10504456959364.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $1$ | |
9.905...992.18t185.a.a | $9$ | $ 2^{6} \cdot 19^{6} \cdot 67^{3}$ | 8.4.10504456959364.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $-1$ | |
9.272...896.18t179.a.a | $9$ | $ 2^{6} \cdot 19^{6} \cdot 67^{6}$ | 8.4.10504456959364.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $1$ | |
12.435...336.24t1503.a.a | $12$ | $ 2^{10} \cdot 19^{6} \cdot 67^{6}$ | 8.4.10504456959364.1 | $(A_4\wr C_2):C_2$ (as 8T45) | $1$ | $0$ |