Normalized defining polynomial
\( x^{12} - 2x^{11} + 6x^{9} - 6x^{8} - 6x^{7} + 14x^{6} - 5x^{5} - 9x^{4} + 10x^{3} - x^{2} - 4x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(854660453125\)
\(\medspace = 5^{6}\cdot 54698269\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.87\) | 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: | $5^{1/2}54698269^{1/2}\approx 16537.573733773646$ | ||
Ramified primes: |
\(5\), \(54698269\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{54698269}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{313}a^{11}+\frac{94}{313}a^{10}-\frac{53}{313}a^{9}-\frac{74}{313}a^{8}+\frac{89}{313}a^{7}+\frac{87}{313}a^{6}-\frac{85}{313}a^{5}-\frac{27}{313}a^{4}-\frac{97}{313}a^{3}+\frac{88}{313}a^{2}-\frac{4}{313}a-\frac{75}{313}$
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)
| |
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: |
$\frac{144}{313}a^{11}-\frac{236}{313}a^{10}-\frac{120}{313}a^{9}+\frac{925}{313}a^{8}-\frac{643}{313}a^{7}-\frac{1244}{313}a^{6}+\frac{1845}{313}a^{5}-\frac{132}{313}a^{4}-\frac{1761}{313}a^{3}+\frac{1091}{313}a^{2}+\frac{50}{313}a-\frac{471}{313}$, $\frac{87}{313}a^{11}-\frac{273}{313}a^{10}+\frac{397}{313}a^{9}+\frac{135}{313}a^{8}-\frac{1021}{313}a^{7}+\frac{996}{313}a^{6}+\frac{743}{313}a^{5}-\frac{2349}{313}a^{4}+\frac{1577}{313}a^{3}+\frac{457}{313}a^{2}-\frac{1287}{313}a+\frac{674}{313}$, $\frac{180}{313}a^{11}-\frac{295}{313}a^{10}-\frac{150}{313}a^{9}+\frac{1078}{313}a^{8}-\frac{569}{313}a^{7}-\frac{1555}{313}a^{6}+\frac{1915}{313}a^{5}+\frac{461}{313}a^{4}-\frac{1810}{313}a^{3}+\frac{816}{313}a^{2}+\frac{532}{313}a-\frac{667}{313}$, $\frac{85}{313}a^{11}-\frac{148}{313}a^{10}-\frac{123}{313}a^{9}+\frac{596}{313}a^{8}-\frac{260}{313}a^{7}-\frac{1056}{313}a^{6}+\frac{1226}{313}a^{5}+\frac{522}{313}a^{4}-\frac{1359}{313}a^{3}+\frac{594}{313}a^{2}+\frac{599}{313}a-\frac{741}{313}$, $\frac{316}{313}a^{11}-\frac{657}{313}a^{10}+\frac{154}{313}a^{9}+\frac{1656}{313}a^{8}-\frac{1924}{313}a^{7}-\frac{1304}{313}a^{6}+\frac{3814}{313}a^{5}-\frac{1959}{313}a^{4}-\frac{1543}{313}a^{3}+\frac{2142}{313}a^{2}-\frac{325}{313}a-\frac{225}{313}$, $\frac{27}{313}a^{11}+\frac{34}{313}a^{10}-\frac{179}{313}a^{9}+\frac{193}{313}a^{8}+\frac{212}{313}a^{7}-\frac{468}{313}a^{6}-\frac{104}{313}a^{5}+\frac{523}{313}a^{4}-\frac{428}{313}a^{3}+\frac{185}{313}a^{2}-\frac{108}{313}a-\frac{147}{313}$, $\frac{266}{313}a^{11}-\frac{662}{313}a^{10}+\frac{300}{313}a^{9}+\frac{1287}{313}a^{8}-\frac{1992}{313}a^{7}-\frac{646}{313}a^{6}+\frac{3369}{313}a^{5}-\frac{2487}{313}a^{4}-\frac{449}{313}a^{3}+\frac{2124}{313}a^{2}-\frac{751}{313}a-\frac{231}{313}$
| 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: | \( 12.1177669565 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 12.1177669565 \cdot 1}{2\cdot\sqrt{854660453125}}\cr\approx \mathstrut & 0.163431306345 \end{aligned}\]
Galois group
$S_6\wr C_2$ (as 12T299):
A non-solvable group of order 1036800 |
The 77 conjugacy class representatives for $S_6\wr C_2$ |
Character table for $S_6\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 20 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Degree 40 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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\)
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(54698269\)
| $\Q_{54698269}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{54698269}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |