Normalized defining polynomial
\( x^{12} - 39 x^{10} - 26 x^{9} + 531 x^{8} + 708 x^{7} - 2653 x^{6} - 5778 x^{5} + 1089 x^{4} + 12320 x^{3} + 13176 x^{2} + 5856 x + 976 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(720795819784500000000\) \(\medspace = 2^{8}\cdot 3^{18}\cdot 5^{9}\cdot 61^{2}\) | 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: | \(54.72\) | 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: | $2\cdot 3^{313/162}5^{3/4}61^{2/3}\approx 865.6110187539238$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{8}+\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{64}a^{10}-\frac{1}{16}a^{9}-\frac{11}{64}a^{8}-\frac{15}{32}a^{7}+\frac{7}{64}a^{6}-\frac{9}{64}a^{4}+\frac{9}{32}a^{3}+\frac{13}{64}a^{2}+\frac{1}{16}a+\frac{1}{16}$, $\frac{1}{512}a^{11}+\frac{1}{256}a^{10}+\frac{29}{512}a^{9}-\frac{7}{16}a^{8}-\frac{109}{512}a^{7}+\frac{53}{256}a^{6}+\frac{55}{512}a^{5}-\frac{9}{128}a^{4}-\frac{71}{512}a^{3}+\frac{9}{256}a^{2}+\frac{55}{128}a-\frac{13}{64}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $\frac{89073}{512}a^{11}-\frac{30843}{256}a^{10}-\frac{3431123}{512}a^{9}+\frac{7533}{64}a^{8}+\frac{47255859}{512}a^{7}+\frac{15168753}{256}a^{6}-\frac{257318073}{512}a^{5}-\frac{84115539}{128}a^{4}+\frac{329998713}{512}a^{3}+\frac{434417661}{256}a^{2}+\frac{142987599}{128}a+\frac{15693111}{64}$, $\frac{6561}{512}a^{11}-\frac{2187}{256}a^{10}-\frac{252963}{512}a^{9}-\frac{243}{64}a^{8}+\frac{3485187}{512}a^{7}+\frac{1160865}{256}a^{6}-\frac{18954153}{512}a^{5}-\frac{6318339}{128}a^{4}+\frac{23993833}{512}a^{3}+\frac{32417901}{256}a^{2}+\frac{10805967}{128}a+\frac{1200407}{64}$, $\frac{40095}{512}a^{11}-\frac{13689}{256}a^{10}-\frac{1545021}{512}a^{9}+\frac{783}{32}a^{8}+\frac{21282285}{512}a^{7}+\frac{6927427}{256}a^{6}-\frac{115837911}{512}a^{5}-\frac{38142891}{128}a^{4}+\frac{147873927}{512}a^{3}+\frac{196508511}{256}a^{2}+\frac{64969209}{128}a+\frac{7159893}{64}$, $\frac{5265}{256}a^{11}-\frac{1809}{128}a^{10}-\frac{202851}{256}a^{9}+\frac{627}{64}a^{8}+\frac{2794075}{256}a^{7}+\frac{903767}{128}a^{6}-\frac{15211257}{256}a^{5}-\frac{312005}{4}a^{4}+\frac{19461217}{256}a^{3}+\frac{25747551}{128}a^{2}+\frac{8494283}{64}a+\frac{934201}{32}$, $\frac{17689}{256}a^{11}-\frac{6151}{128}a^{10}-\frac{681291}{256}a^{9}+\frac{433}{8}a^{8}+\frac{9382331}{256}a^{7}+\frac{2999821}{128}a^{6}-\frac{51089537}{256}a^{5}-\frac{16669825}{64}a^{4}+\frac{65570097}{256}a^{3}+\frac{86149185}{128}a^{2}+\frac{28339511}{64}a+\frac{3109603}{32}$, $\frac{21879}{512}a^{11}-\frac{6961}{256}a^{10}-\frac{844485}{512}a^{9}-\frac{1963}{32}a^{8}+\frac{11640117}{512}a^{7}+\frac{4040971}{256}a^{6}-\frac{63220463}{512}a^{5}-\frac{21547387}{128}a^{4}+\frac{78855903}{512}a^{3}+\frac{109748647}{256}a^{2}+\frac{37072417}{128}a+\frac{4167277}{64}$, $\frac{35265}{512}a^{11}-\frac{12375}{256}a^{10}-\frac{1357987}{512}a^{9}+\frac{2267}{32}a^{8}+\frac{18700851}{512}a^{7}+\frac{5920461}{256}a^{6}-\frac{101874505}{512}a^{5}-\frac{33062845}{128}a^{4}+\frac{131253081}{512}a^{3}+\frac{171169425}{256}a^{2}+\frac{56081911}{128}a+\frac{6128299}{64}$, $\frac{3857}{128}a^{11}-\frac{607}{32}a^{10}-\frac{148939}{128}a^{9}-\frac{3227}{64}a^{8}+\frac{2053783}{128}a^{7}+\frac{179551}{16}a^{6}-\frac{11159209}{128}a^{5}-\frac{7628347}{64}a^{4}+\frac{13929629}{128}a^{3}+\frac{9702443}{32}a^{2}+\frac{6542957}{32}a+45807$, $\frac{35109}{256}a^{11}-\frac{12073}{128}a^{10}-\frac{1352639}{256}a^{9}+\frac{4357}{64}a^{8}+\frac{18630695}{256}a^{7}+\frac{6022087}{128}a^{6}-\frac{101425069}{256}a^{5}-\frac{16638239}{32}a^{4}+\frac{129764517}{256}a^{3}+\frac{171647319}{128}a^{2}+\frac{56629023}{64}a+\frac{6228841}{32}$, $\frac{51853}{256}a^{11}-\frac{17993}{128}a^{10}-\frac{1997271}{256}a^{9}+\frac{9473}{64}a^{8}+\frac{27506847}{256}a^{7}+\frac{8811479}{128}a^{6}-\frac{149785653}{256}a^{5}-\frac{24457273}{32}a^{4}+\frac{192195405}{256}a^{3}+\frac{252709783}{128}a^{2}+\frac{83139199}{64}a+\frac{9121281}{32}$, $\frac{230153}{256}a^{11}-\frac{79227}{128}a^{10}-\frac{8866907}{256}a^{9}+\frac{15087}{32}a^{8}+\frac{122129307}{256}a^{7}+\frac{39432257}{128}a^{6}-\frac{664907665}{256}a^{5}-\frac{218012351}{64}a^{4}+\frac{851106929}{256}a^{3}+\frac{1124780189}{128}a^{2}+\frac{370895239}{64}a+\frac{40774919}{32}$ (assuming GRH) | 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: | \( 3902244.46053 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 3902244.46053 \cdot 1}{2\cdot\sqrt{720795819784500000000}}\cr\approx \mathstrut & 0.297672212491 \end{aligned}\] (assuming GRH)
Galois group
$C_3^3:(C_4\times S_3)$ (as 12T170):
A solvable group of order 648 |
The 30 conjugacy class representatives for $C_3^3:(C_4\times S_3)$ |
Character table for $C_3^3:(C_4\times S_3)$ is not computed |
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
Degree 12 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 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 | R | R | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
\(3\) | 3.12.18.8 | $x^{12} + 12 x^{11} + 42 x^{10} + 36 x^{9} + 9 x^{8} - 42 x^{6} - 252 x^{5} - 126 x^{4} + 882$ | $6$ | $2$ | $18$ | 12T170 | $[3/2, 3/2, 2, 2]_{2}^{4}$ |
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(61\) | 61.3.2.1 | $x^{3} + 61$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |