Normalized defining polynomial
\( x^{12} - 6 x^{11} + 18 x^{10} - 32 x^{9} + 44 x^{8} - 64 x^{7} + 104 x^{6} - 116 x^{5} + 68 x^{4} + \cdots + 4 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(44121902874624\) \(\medspace = 2^{16}\cdot 3^{6}\cdot 31^{4}\) | 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: | \(13.71\) | 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: | $2^{4/3}3^{1/2}31^{2/3}\approx 43.07002106143874$ | ||
Ramified primes: | \(2\), \(3\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{13724}a^{11}-\frac{693}{6862}a^{10}+\frac{1631}{13724}a^{9}-\frac{19}{3431}a^{8}+\frac{997}{6862}a^{7}-\frac{61}{6862}a^{6}-\frac{1543}{6862}a^{5}+\frac{1031}{3431}a^{4}-\frac{1227}{6862}a^{3}-\frac{829}{3431}a^{2}+\frac{1499}{3431}a+\frac{271}{3431}$
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: | \( \frac{232}{3431} a^{11} - \frac{6445}{13724} a^{10} + \frac{21083}{13724} a^{9} - \frac{39649}{13724} a^{8} + \frac{13147}{3431} a^{7} - \frac{18011}{3431} a^{6} + \frac{60581}{6862} a^{5} - \frac{73013}{6862} a^{4} + \frac{38177}{6862} a^{3} + \frac{8757}{6862} a^{2} - \frac{1914}{3431} a - \frac{2406}{3431} \) (order $12$) | 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: | $\frac{232}{3431}a^{11}-\frac{6445}{13724}a^{10}+\frac{21083}{13724}a^{9}-\frac{39649}{13724}a^{8}+\frac{13147}{3431}a^{7}-\frac{18011}{3431}a^{6}+\frac{60581}{6862}a^{5}-\frac{73013}{6862}a^{4}+\frac{38177}{6862}a^{3}+\frac{8757}{6862}a^{2}-\frac{1914}{3431}a-\frac{5837}{3431}$, $\frac{215}{3431}a^{11}-\frac{2417}{6862}a^{10}+\frac{13105}{13724}a^{9}-\frac{20757}{13724}a^{8}+\frac{6697}{3431}a^{7}-\frac{21581}{6862}a^{6}+\frac{35127}{6862}a^{5}-\frac{15693}{3431}a^{4}+\frac{11821}{6862}a^{3}-\frac{2015}{6862}a^{2}+\frac{2515}{3431}a+\frac{3183}{3431}$, $\frac{215}{3431}a^{11}-\frac{2417}{6862}a^{10}+\frac{13105}{13724}a^{9}-\frac{20757}{13724}a^{8}+\frac{6697}{3431}a^{7}-\frac{21581}{6862}a^{6}+\frac{35127}{6862}a^{5}-\frac{15693}{3431}a^{4}+\frac{11821}{6862}a^{3}-\frac{2015}{6862}a^{2}+\frac{2515}{3431}a-\frac{248}{3431}$, $\frac{3}{73}a^{11}-\frac{61}{292}a^{10}+\frac{77}{146}a^{9}-\frac{91}{146}a^{8}+\frac{65}{146}a^{7}-\frac{75}{146}a^{6}+\frac{86}{73}a^{5}-\frac{3}{146}a^{4}-\frac{208}{73}a^{3}+\frac{345}{73}a^{2}-\frac{189}{73}a+\frac{40}{73}$, $\frac{33}{292}a^{11}-\frac{93}{146}a^{10}+\frac{533}{292}a^{9}-\frac{451}{146}a^{8}+\frac{635}{146}a^{7}-\frac{991}{146}a^{6}+\frac{1641}{146}a^{5}-\frac{871}{73}a^{4}+\frac{1119}{146}a^{3}-\frac{274}{73}a^{2}+\frac{338}{73}a-\frac{109}{73}$ | 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: | \( 471.20242897812045 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 471.20242897812045 \cdot 1}{12\cdot\sqrt{44121902874624}}\cr\approx \mathstrut & 0.363729367561760 \end{aligned}\]
Galois group
$C_6\times S_3$ (as 12T18):
A solvable group of order 36 |
The 18 conjugacy class representatives for $C_6\times S_3$ |
Character table for $C_6\times S_3$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 6.0.415152.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 36 |
Degree 18 siblings: | 18.6.260106685170802123480454135808.1, 18.0.9633580932251930499276079104.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.16.13 | $x^{12} + 10 x^{11} + 47 x^{10} + 144 x^{9} + 330 x^{8} + 578 x^{7} + 785 x^{6} + 830 x^{5} + 530 x^{4} - 64 x^{3} - 189 x^{2} - 30 x + 25$ | $6$ | $2$ | $16$ | $D_6$ | $[2]_{3}^{2}$ |
\(3\) | 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(31\) | 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.6.4.1 | $x^{6} + 87 x^{5} + 2532 x^{4} + 24973 x^{3} + 10293 x^{2} + 78438 x + 748956$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.372.6t1.b.a | $1$ | $ 2^{2} \cdot 3 \cdot 31 $ | 6.6.1595844288.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.124.6t1.a.a | $1$ | $ 2^{2} \cdot 31 $ | 6.0.59105344.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.124.6t1.a.b | $1$ | $ 2^{2} \cdot 31 $ | 6.0.59105344.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.93.6t1.a.a | $1$ | $ 3 \cdot 31 $ | 6.0.24935067.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.372.6t1.b.b | $1$ | $ 2^{2} \cdot 3 \cdot 31 $ | 6.6.1595844288.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.93.6t1.a.b | $1$ | $ 3 \cdot 31 $ | 6.0.24935067.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.31.3t1.a.a | $1$ | $ 31 $ | 3.3.961.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.31.3t1.a.b | $1$ | $ 31 $ | 3.3.961.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
2.11532.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 31^{2}$ | 3.1.11532.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.46128.6t3.a.a | $2$ | $ 2^{4} \cdot 3 \cdot 31^{2}$ | 6.0.2127792384.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.372.6t5.b.a | $2$ | $ 2^{2} \cdot 3 \cdot 31 $ | 6.0.415152.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.372.6t5.b.b | $2$ | $ 2^{2} \cdot 3 \cdot 31 $ | 6.0.415152.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.1488.12t18.a.a | $2$ | $ 2^{4} \cdot 3 \cdot 31 $ | 12.0.44121902874624.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.1488.12t18.a.b | $2$ | $ 2^{4} \cdot 3 \cdot 31 $ | 12.0.44121902874624.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |