Normalized defining polynomial
\( x^{12} - 12x^{10} + 18x^{8} + 124x^{6} + 81x^{4} + 144x^{2} + 32 \)
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: | \(369768517790072832\) \(\medspace = 2^{33}\cdot 3^{16}\) | 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: | \(29.11\) | 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: | $2^{11/4}3^{16/9}\approx 47.429446013156884$ | ||
Ramified primes: | \(2\), \(3\) | 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: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{8}+\frac{3}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{15088}a^{10}+\frac{39}{656}a^{8}-\frac{1247}{15088}a^{6}-\frac{1799}{15088}a^{4}+\frac{931}{3772}a^{2}-\frac{1}{2}a+\frac{111}{943}$, $\frac{1}{15088}a^{11}+\frac{39}{656}a^{9}+\frac{639}{15088}a^{7}+\frac{1973}{15088}a^{5}-\frac{1}{2}a^{4}+\frac{919}{7544}a^{3}+\frac{111}{943}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( -1 \) (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: | $\frac{21}{943}a^{10}-\frac{45}{164}a^{8}+\frac{1811}{3772}a^{6}+\frac{10137}{3772}a^{4}+\frac{4455}{3772}a^{2}+\frac{519}{943}$, $\frac{57}{656}a^{11}-\frac{81}{15088}a^{10}-\frac{695}{656}a^{9}+\frac{39}{656}a^{8}+\frac{1163}{656}a^{7}-\frac{837}{15088}a^{6}+\frac{6845}{656}a^{5}-\frac{10819}{15088}a^{4}+\frac{1625}{328}a^{3}-\frac{1857}{3772}a^{2}+\frac{887}{82}a-\frac{1447}{943}$, $\frac{81}{7544}a^{10}-\frac{39}{328}a^{8}+\frac{837}{7544}a^{6}+\frac{10819}{7544}a^{4}+\frac{1857}{1886}a^{2}+\frac{65}{943}$, $\frac{11267}{15088}a^{11}-\frac{11563}{15088}a^{10}-\frac{6995}{656}a^{9}+\frac{7177}{656}a^{8}+\frac{568437}{15088}a^{7}-\frac{582183}{15088}a^{6}+\frac{110783}{15088}a^{5}-\frac{123333}{15088}a^{4}+\frac{328397}{7544}a^{3}-\frac{169605}{3772}a^{2}+\frac{20241}{1886}a-\frac{10443}{943}$, $\frac{11267}{15088}a^{11}+\frac{11563}{15088}a^{10}-\frac{6995}{656}a^{9}-\frac{7177}{656}a^{8}+\frac{568437}{15088}a^{7}+\frac{582183}{15088}a^{6}+\frac{110783}{15088}a^{5}+\frac{123333}{15088}a^{4}+\frac{328397}{7544}a^{3}+\frac{169605}{3772}a^{2}+\frac{20241}{1886}a+\frac{10443}{943}$ | 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: | \( 50344.2311125 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 50344.2311125 \cdot 1}{2\cdot\sqrt{369768517790072832}}\cr\approx \mathstrut & 2.54702941592 \end{aligned}\]
Galois group
$C_3^2:C_4$ (as 12T17):
A solvable group of order 36 |
The 6 conjugacy class representatives for $(C_3\times C_3):C_4$ |
Character table for $(C_3\times C_3):C_4$ |
Intermediate fields
\(\Q(\sqrt{2}) \), 4.0.2048.2, 6.2.107495424.5 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 36 |
Degree 6 siblings: | 6.2.107495424.5, 6.2.107495424.2 |
Degree 9 sibling: | 9.1.180551034077184.1 |
Degree 12 sibling: | 12.0.369768517790072832.60 |
Degree 18 sibling: | deg 18 |
Minimal sibling: | 6.2.107495424.5 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.1.0.1}{1} }^{12}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.11.2 | $x^{4} + 4 x^{2} + 2$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ |
2.4.11.2 | $x^{4} + 4 x^{2} + 2$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ | |
2.4.11.2 | $x^{4} + 4 x^{2} + 2$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ | |
\(3\) | 3.12.16.37 | $x^{12} - 6 x^{11} + 60 x^{10} - 10 x^{9} + 603 x^{8} + 3924 x^{7} + 16740 x^{6} + 36078 x^{5} + 108120 x^{4} + 201600 x^{3} + 381102 x^{2} + 225588 x + 200729$ | $3$ | $4$ | $16$ | $(C_3\times C_3):C_4$ | $[2, 2]^{4}$ |