Normalized defining polynomial
\( x^{12} - 2x^{11} - 2x^{10} + 6x^{9} - x^{8} - 16x^{7} + 6x^{6} + 40x^{5} - 7x^{4} - 30x^{3} + 4x^{2} + 6x - 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(241875776000000\) \(\medspace = 2^{12}\cdot 5^{6}\cdot 19^{4}\cdot 29\) | 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: | \(15.80\) | 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: | $2^{31/24}5^{1/2}19^{2/3}29^{1/2}\approx 209.9021141342055$ | ||
Ramified primes: | \(2\), \(5\), \(19\), \(29\) | 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: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{1522}a^{11}-\frac{67}{1522}a^{10}-\frac{213}{1522}a^{9}+\frac{153}{1522}a^{8}-\frac{53}{1522}a^{7}-\frac{188}{761}a^{6}-\frac{667}{1522}a^{5}+\frac{9}{761}a^{4}-\frac{208}{761}a^{3}+\frac{375}{1522}a^{2}+\frac{371}{761}a-\frac{281}{1522}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{825}{1522}a^{11}+\frac{622}{761}a^{10}+\frac{2217}{1522}a^{9}-\frac{1852}{761}a^{8}-\frac{587}{761}a^{7}+\frac{12649}{1522}a^{6}+\frac{797}{761}a^{5}-\frac{32353}{1522}a^{4}-\frac{10665}{1522}a^{3}+\frac{19377}{1522}a^{2}+\frac{6543}{1522}a-\frac{4085}{1522}$, $\frac{177}{1522}a^{11}+\frac{317}{1522}a^{10}-\frac{967}{761}a^{9}+\frac{223}{761}a^{8}+\frac{1778}{761}a^{7}-\frac{4911}{1522}a^{6}-\frac{3857}{761}a^{5}+\frac{13079}{1522}a^{4}+\frac{21493}{1522}a^{3}-\frac{7526}{761}a^{2}-\frac{5106}{761}a+\frac{3533}{1522}$, $\frac{177}{1522}a^{11}-\frac{317}{1522}a^{10}+\frac{967}{761}a^{9}-\frac{223}{761}a^{8}-\frac{1778}{761}a^{7}+\frac{4911}{1522}a^{6}+\frac{3857}{761}a^{5}-\frac{13079}{1522}a^{4}-\frac{21493}{1522}a^{3}+\frac{7526}{761}a^{2}+\frac{5106}{761}a-\frac{5055}{1522}$, $a$, $\frac{420}{761}a^{11}-\frac{744}{761}a^{10}-\frac{1607}{1522}a^{9}+\frac{1858}{761}a^{8}-\frac{191}{761}a^{7}-\frac{5720}{761}a^{6}+\frac{669}{761}a^{5}+\frac{14409}{761}a^{4}+\frac{1832}{761}a^{3}-\frac{5354}{761}a^{2}-\frac{3023}{1522}a-\frac{65}{761}$, $\frac{31}{1522}a^{11}-\frac{103}{761}a^{10}+\frac{515}{1522}a^{9}+\frac{292}{761}a^{8}-\frac{1401}{1522}a^{7}+\frac{241}{1522}a^{6}+\frac{3935}{1522}a^{5}-\frac{1319}{1522}a^{4}-\frac{5728}{761}a^{3}-\frac{2493}{1522}a^{2}+\frac{3719}{761}a+\frac{2623}{1522}$, $\frac{1309}{1522}a^{11}+\frac{2471}{1522}a^{10}+\frac{1287}{761}a^{9}-\frac{6983}{1522}a^{8}+\frac{887}{1522}a^{7}+\frac{19603}{1522}a^{6}-\frac{5091}{1522}a^{5}-\frac{48675}{1522}a^{4}+\frac{1356}{761}a^{3}+\frac{15205}{761}a^{2}+\frac{519}{1522}a-\frac{2150}{761}$, $\frac{85}{1522}a^{11}-\frac{577}{761}a^{10}+\frac{2885}{1522}a^{9}+\frac{727}{761}a^{8}-\frac{3455}{761}a^{7}+\frac{3803}{1522}a^{6}+\frac{8942}{761}a^{5}-\frac{11423}{1522}a^{4}-\frac{44545}{1522}a^{3}+\frac{12263}{1522}a^{2}+\frac{21401}{1522}a-\frac{3511}{1522}$, $\frac{729}{1522}a^{11}-\frac{1661}{1522}a^{10}-\frac{397}{761}a^{9}+\frac{2118}{761}a^{8}-\frac{2109}{1522}a^{7}-\frac{10037}{1522}a^{6}+\frac{6885}{1522}a^{5}+\frac{24537}{1522}a^{4}-\frac{5520}{761}a^{3}-\frac{6761}{761}a^{2}+\frac{2891}{1522}a-\frac{901}{1522}$ | 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: | \( 662.2910309313443 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 662.2910309313443 \cdot 1}{2\cdot\sqrt{241875776000000}}\cr\approx \mathstrut & 0.215190151928509 \end{aligned}\]
Galois group
$A_4^2:D_4$ (as 12T208):
A solvable group of order 1152 |
The 44 conjugacy class representatives for $A_4^2:D_4$ |
Character table for $A_4^2:D_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 6.6.722000.1 |
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 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | 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 | ${\href{/padicField/3.12.0.1}{12} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.29 | $x^{12} + 4 x^{8} + 4 x^{7} - 2 x^{6} + 4 x^{4} + 8 x^{3} - 4 x + 4$ | $6$ | $2$ | $12$ | 12T159 | $[4/3, 4/3, 4/3, 4/3]_{3}^{12}$ |
\(5\) | 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(19\) | 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
19.6.4.2 | $x^{6} - 342 x^{3} + 722$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.6.0.1 | $x^{6} + x^{4} + 25 x^{3} + 17 x^{2} + 13 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |