Normalized defining polynomial
\( x^{12} - 4 x^{11} - 2 x^{10} + 20 x^{9} + 15 x^{8} - 40 x^{7} - 60 x^{6} + 80 x^{5} + 270 x^{4} + 288 x^{3} + 192 x^{2} + 128 x + 64 \)
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: | \(49519263525896192\) \(\medspace = 2^{33}\cdot 7^{8}\) | 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: | \(24.62\) | 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: | $2^{11/4}7^{2/3}\approx 24.616776431006123$ | ||
Ramified primes: | \(2\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{90700304}a^{11}-\frac{915961}{45350152}a^{10}-\frac{1518341}{45350152}a^{9}+\frac{1998947}{22675076}a^{8}-\frac{5041965}{90700304}a^{7}-\frac{10847043}{45350152}a^{6}-\frac{345697}{11337538}a^{5}-\frac{1820374}{5668769}a^{4}+\frac{5236523}{45350152}a^{3}+\frac{418721}{1333828}a^{2}+\frac{5139753}{11337538}a+\frac{1929770}{5668769}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}\times C_{3}$, which has order $9$
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: | $\frac{55457}{2667656}a^{11}-\frac{221449}{2667656}a^{10}-\frac{75939}{1333828}a^{9}+\frac{650113}{1333828}a^{8}+\frac{778291}{2667656}a^{7}-\frac{3081319}{2667656}a^{6}-\frac{750827}{666914}a^{5}+\frac{1501473}{666914}a^{4}+\frac{7492991}{1333828}a^{3}+\frac{6558133}{1333828}a^{2}+\frac{573462}{333457}a+\frac{130991}{333457}$, $\frac{1484913}{90700304}a^{11}-\frac{3312685}{45350152}a^{10}+\frac{1192423}{45350152}a^{9}+\frac{4237907}{22675076}a^{8}+\frac{22569787}{90700304}a^{7}-\frac{15051799}{45350152}a^{6}-\frac{11598873}{11337538}a^{5}+\frac{4461267}{5668769}a^{4}+\frac{180066035}{45350152}a^{3}+\frac{8337041}{1333828}a^{2}+\frac{52278719}{11337538}a+\frac{6174355}{5668769}$, $\frac{891139}{90700304}a^{11}-\frac{2521269}{45350152}a^{10}+\frac{2917135}{45350152}a^{9}+\frac{666595}{5668769}a^{8}-\frac{2663659}{90700304}a^{7}-\frac{19553785}{45350152}a^{6}-\frac{496347}{11337538}a^{5}+\frac{11036105}{11337538}a^{4}+\frac{52604277}{45350152}a^{3}+\frac{230219}{1333828}a^{2}-\frac{6290415}{11337538}a-\frac{1462117}{5668769}$, $\frac{245257}{45350152}a^{11}-\frac{2069321}{45350152}a^{10}+\frac{1672871}{11337538}a^{9}-\frac{2073817}{11337538}a^{8}-\frac{3277883}{45350152}a^{7}+\frac{17776839}{45350152}a^{6}+\frac{668839}{11337538}a^{5}-\frac{2783201}{5668769}a^{4}-\frac{11045691}{22675076}a^{3}+\frac{2009127}{1333828}a^{2}+\frac{13245298}{5668769}a-\frac{8852147}{5668769}$, $\frac{1670881}{45350152}a^{11}-\frac{9485273}{45350152}a^{10}+\frac{2702997}{11337538}a^{9}+\frac{12766549}{22675076}a^{8}-\frac{29859809}{45350152}a^{7}-\frac{46685543}{45350152}a^{6}+\frac{8341561}{22675076}a^{5}+\frac{42466891}{11337538}a^{4}+\frac{65910947}{22675076}a^{3}+\frac{1325093}{1333828}a^{2}+\frac{22402917}{11337538}a+\frac{758957}{5668769}$ | 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: | \( 2643.94789139 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2643.94789139 \cdot 9}{2\cdot\sqrt{49519263525896192}}\cr\approx \mathstrut & 3.28970933319 \end{aligned}\]
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, 6.2.39337984.1 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: | 6.2.802816.1, 6.2.39337984.1 |
Degree 9 sibling: | 9.1.493455671296.1 |
Degree 12 sibling: | 12.0.20624432955392.2 |
Degree 18 sibling: | 18.2.1947987996273487986556928.1 |
Minimal sibling: | 6.2.802816.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 | $x^{4} + 4 x^{2} + 2$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ | |
\(7\) | 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |