Normalized defining polynomial
\( x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{9} + 6 x^{8} - 10 x^{7} + 10 x^{6} - 12 x^{5} + 17 x^{4} - 14 x^{3} + \cdots + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(73358639104\) \(\medspace = 2^{18}\cdot 23^{4}\) | 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: | \(8.04\) | 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: | $2^{7/4}23^{1/2}\approx 16.131190144457708$ | ||
Ramified primes: | \(2\), \(23\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{79}a^{11}-\frac{14}{79}a^{10}+\frac{12}{79}a^{9}+\frac{12}{79}a^{8}+\frac{20}{79}a^{7}-\frac{13}{79}a^{6}+\frac{8}{79}a^{5}-\frac{29}{79}a^{4}-\frac{30}{79}a^{3}+\frac{30}{79}a^{2}-\frac{36}{79}a+\frac{33}{79}$
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: | \( \frac{3985}{79} a^{11} - \frac{5546}{79} a^{10} + \frac{4607}{79} a^{9} - \frac{5189}{79} a^{8} + \frac{20766}{79} a^{7} - \frac{27236}{79} a^{6} + \frac{23348}{79} a^{5} - \frac{33721}{79} a^{4} + \frac{47298}{79} a^{3} - \frac{27153}{79} a^{2} + \frac{15567}{79} a - \frac{6587}{79} \) (order $4$) | 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: | $\frac{1485}{79}a^{11}-\frac{2067}{79}a^{10}+\frac{1704}{79}a^{9}-\frac{1930}{79}a^{8}+\frac{7738}{79}a^{7}-\frac{10141}{79}a^{6}+\frac{8641}{79}a^{5}-\frac{12571}{79}a^{4}+\frac{17623}{79}a^{3}-\frac{10039}{79}a^{2}+\frac{5711}{79}a-\frac{2503}{79}$, $\frac{1620}{79}a^{11}-\frac{2298}{79}a^{10}+\frac{1902}{79}a^{9}-\frac{2127}{79}a^{8}+\frac{8463}{79}a^{7}-\frac{11264}{79}a^{6}+\frac{9642}{79}a^{5}-\frac{13800}{79}a^{4}+\frac{19419}{79}a^{3}-\frac{11282}{79}a^{2}+\frac{6302}{79}a-\frac{2709}{79}$, $\frac{2025}{79}a^{11}-\frac{2754}{79}a^{10}+\frac{2259}{79}a^{9}-\frac{2560}{79}a^{8}+\frac{10480}{79}a^{7}-\frac{13527}{79}a^{6}+\frac{11460}{79}a^{5}-\frac{16776}{79}a^{4}+\frac{23543}{79}a^{3}-\frac{13115}{79}a^{2}+\frac{7522}{79}a-\frac{3169}{79}$, $\frac{2610}{79}a^{11}-\frac{3597}{79}a^{10}+\frac{2959}{79}a^{9}-\frac{3361}{79}a^{8}+\frac{13569}{79}a^{7}-\frac{17656}{79}a^{6}+\frac{15034}{79}a^{5}-\frac{21891}{79}a^{4}+\frac{30720}{79}a^{3}-\frac{17369}{79}a^{2}+\frac{10004}{79}a-\frac{4246}{79}$, $\frac{1375}{79}a^{11}-\frac{1949}{79}a^{10}+\frac{1648}{79}a^{9}-\frac{1828}{79}a^{8}+\frac{7197}{79}a^{7}-\frac{9580}{79}a^{6}+\frac{8314}{79}a^{5}-\frac{11830}{79}a^{4}+\frac{16578}{79}a^{3}-\frac{9784}{79}a^{2}+\frac{5563}{79}a-\frac{2262}{79}$ | 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: | \( 3.42242551816 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3.42242551816 \cdot 1}{4\cdot\sqrt{73358639104}}\cr\approx \mathstrut & 0.194369264472 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T23):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2 \times S_4$ |
Character table for $C_2 \times S_4$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 3.1.23.1, 6.0.33856.1, 6.0.33856.2, 6.2.33856.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.778688.4, 6.0.33856.1 |
Degree 8 siblings: | 8.0.8667136.1, 8.4.4584914944.1 |
Degree 12 siblings: | 12.2.1687248699392.2, 12.0.606355001344.2, 12.4.38806720086016.2, 12.0.38806720086016.1, 12.0.38806720086016.3 |
Degree 16 sibling: | deg 16 |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.33856.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.51 | $x^{12} + 2 x^{11} + 16 x^{10} + 44 x^{9} + 18 x^{8} - 8 x^{7} + 24 x^{6} + 40 x^{5} + 20 x^{4} + 8 x^{3} + 8$ | $4$ | $3$ | $18$ | $A_4 \times C_2$ | $[2, 2, 2]^{3}$ |
\(23\) | 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |