Normalized defining polynomial
\( x^{12} - x^{11} - 17 x^{10} + 8 x^{9} + 79 x^{8} - 32 x^{7} - 126 x^{6} + 37 x^{5} + 81 x^{4} - 15 x^{3} + \cdots + 1 \)
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: | \(2196839556078125\) \(\medspace = 5^{6}\cdot 7^{8}\cdot 29^{3}\) | 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: | \(18.99\) | 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: | $5^{1/2}7^{2/3}29^{1/2}\approx 44.06387579995631$ | ||
Ramified primes: | \(5\), \(7\), \(29\) | 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: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13}a^{10}+\frac{1}{13}a^{9}+\frac{3}{13}a^{8}+\frac{6}{13}a^{7}+\frac{2}{13}a^{6}+\frac{2}{13}a^{5}+\frac{5}{13}a^{4}+\frac{5}{13}a^{3}-\frac{1}{13}a^{2}-\frac{5}{13}a+\frac{5}{13}$, $\frac{1}{533}a^{11}+\frac{19}{533}a^{10}-\frac{252}{533}a^{9}+\frac{216}{533}a^{8}-\frac{111}{533}a^{7}-\frac{79}{533}a^{6}+\frac{262}{533}a^{5}-\frac{217}{533}a^{4}+\frac{128}{533}a^{3}+\frac{3}{533}a^{2}+\frac{3}{13}a+\frac{207}{533}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{22}{13}a^{11}-\frac{33}{13}a^{10}-\frac{353}{13}a^{9}+\frac{344}{13}a^{8}+115a^{7}-\frac{1353}{13}a^{6}-138a^{5}+\frac{1330}{13}a^{4}+\frac{769}{13}a^{3}-\frac{367}{13}a^{2}-\frac{83}{13}a+\frac{11}{13}$, $\frac{315}{533}a^{11}-\frac{288}{533}a^{10}-\frac{5170}{533}a^{9}+\frac{1784}{533}a^{8}+\frac{21738}{533}a^{7}-\frac{5451}{533}a^{6}-\frac{27022}{533}a^{5}-\frac{582}{533}a^{4}+\frac{12686}{533}a^{3}+\frac{2954}{533}a^{2}-\frac{58}{13}a-\frac{272}{533}$, $\frac{272}{533}a^{11}+\frac{43}{533}a^{10}-\frac{4912}{533}a^{9}-\frac{2994}{533}a^{8}+\frac{23272}{533}a^{7}+\frac{13034}{533}a^{6}-\frac{39723}{533}a^{5}-\frac{16958}{533}a^{4}+\frac{1650}{41}a^{3}+\frac{662}{41}a^{2}-\frac{54}{13}a-\frac{1834}{533}$, $\frac{424}{533}a^{11}-\frac{1046}{533}a^{10}-\frac{6152}{533}a^{9}+\frac{13110}{533}a^{8}+\frac{21980}{533}a^{7}-\frac{52766}{533}a^{6}-\frac{8919}{533}a^{5}+\frac{54362}{533}a^{4}-\frac{474}{41}a^{3}-\frac{1334}{41}a^{2}+\frac{42}{13}a+\frac{684}{533}$, $\frac{315}{533}a^{11}-\frac{288}{533}a^{10}-\frac{5170}{533}a^{9}+\frac{1784}{533}a^{8}+\frac{21738}{533}a^{7}-\frac{5451}{533}a^{6}-\frac{27022}{533}a^{5}-\frac{582}{533}a^{4}+\frac{12686}{533}a^{3}+\frac{2954}{533}a^{2}-\frac{58}{13}a+\frac{261}{533}$, $\frac{171}{533}a^{11}+\frac{10}{533}a^{10}-\frac{2625}{533}a^{9}-\frac{2096}{533}a^{8}+\frac{571}{41}a^{7}+\frac{10927}{533}a^{6}+\frac{242}{41}a^{5}-\frac{23987}{533}a^{4}-\frac{13495}{533}a^{3}+\frac{14945}{533}a^{2}+\frac{115}{13}a-\frac{2651}{533}$, $\frac{294}{533}a^{11}-\frac{564}{533}a^{10}-\frac{4552}{533}a^{9}+\frac{6678}{533}a^{8}+\frac{18411}{533}a^{7}-\frac{28064}{533}a^{6}-\frac{18953}{533}a^{5}+\frac{33372}{533}a^{4}+\frac{7415}{533}a^{3}-\frac{12689}{533}a^{2}-\frac{32}{13}a+\frac{61}{41}$, $\frac{628}{533}a^{11}-\frac{1188}{533}a^{10}-\frac{9344}{533}a^{9}+\frac{13140}{533}a^{8}+\frac{33325}{533}a^{7}-\frac{48669}{533}a^{6}-\frac{15741}{533}a^{5}+\frac{33177}{533}a^{4}-\frac{6536}{533}a^{3}-\frac{986}{533}a^{2}+5a-\frac{1163}{533}$, $\frac{1205}{533}a^{11}-\frac{87}{41}a^{10}-\frac{20145}{533}a^{9}+\frac{8048}{533}a^{8}+\frac{88793}{533}a^{7}-\frac{2368}{41}a^{6}-\frac{123031}{533}a^{5}+\frac{2133}{41}a^{4}+\frac{55430}{533}a^{3}-\frac{4872}{533}a^{2}-\frac{59}{13}a-\frac{214}{533}$, $\frac{99}{533}a^{11}+\frac{241}{533}a^{10}-\frac{2070}{533}a^{9}-\frac{4856}{533}a^{8}+\frac{11151}{533}a^{7}+\frac{21945}{533}a^{6}-\frac{24779}{533}a^{5}-\frac{31282}{533}a^{4}+\frac{1410}{41}a^{3}+\frac{1256}{41}a^{2}-\frac{88}{13}a-\frac{2631}{533}$, $\frac{389}{533}a^{11}-\frac{440}{533}a^{10}-\frac{476}{41}a^{9}+\frac{3500}{533}a^{8}+\frac{24430}{533}a^{7}-\frac{12281}{533}a^{6}-\frac{24075}{533}a^{5}+\frac{5951}{533}a^{4}+\frac{6373}{533}a^{3}-\frac{596}{533}a^{2}+\frac{3}{13}a+\frac{327}{533}$ | 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: | \( 3668.73449811 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 3668.73449811 \cdot 1}{2\cdot\sqrt{2196839556078125}}\cr\approx \mathstrut & 0.160304999066 \end{aligned}\]
Galois group
$C_3\times D_4$ (as 12T14):
A solvable group of order 24 |
The 15 conjugacy class representatives for $D_4 \times C_3$ |
Character table for $D_4 \times C_3$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.4.725.1, 6.6.300125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 12 sibling: | deg 12 |
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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.12.0.1}{12} }$ | R | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(7\) | 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(29\) | 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |