Normalized defining polynomial
\( x^{12} - 2x^{11} + 2x^{10} + 9x^{8} - 24x^{7} + 30x^{6} - 4x^{5} + 6x^{4} - 46x^{3} + 98x^{2} - 70x + 25 \)
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: | \(38806720086016\) \(\medspace = 2^{18}\cdot 23^{6}\) | 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: | \(13.56\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{23}a^{10}+\frac{8}{23}a^{9}+\frac{9}{23}a^{8}-\frac{11}{23}a^{7}+\frac{1}{23}a^{6}+\frac{7}{23}a^{5}+\frac{4}{23}a^{4}+\frac{8}{23}a^{3}+\frac{1}{23}a^{2}+\frac{1}{23}a-\frac{11}{23}$, $\frac{1}{191705}a^{11}-\frac{1197}{191705}a^{10}+\frac{71812}{191705}a^{9}-\frac{564}{1667}a^{8}-\frac{91141}{191705}a^{7}+\frac{16696}{191705}a^{6}-\frac{6208}{38341}a^{5}-\frac{22959}{191705}a^{4}-\frac{44484}{191705}a^{3}-\frac{77311}{191705}a^{2}-\frac{31737}{191705}a-\frac{9723}{38341}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{921}{8335} a^{11} + \frac{2217}{8335} a^{10} - \frac{627}{8335} a^{9} - \frac{177}{1667} a^{8} - \frac{9259}{8335} a^{7} + \frac{26064}{8335} a^{6} - \frac{3576}{1667} a^{5} - \frac{8991}{8335} a^{4} - \frac{5096}{8335} a^{3} + \frac{64206}{8335} a^{2} - \frac{76083}{8335} a + \frac{6427}{1667} \) (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: | $\frac{5584}{191705}a^{11}+\frac{8957}{191705}a^{10}+\frac{9693}{191705}a^{9}-\frac{1164}{38341}a^{8}+\frac{37096}{191705}a^{7}+\frac{45164}{191705}a^{6}+\frac{9795}{38341}a^{5}-\frac{19091}{191705}a^{4}+\frac{109369}{191705}a^{3}+\frac{190071}{191705}a^{2}+\frac{91047}{191705}a-\frac{4043}{38341}$, $\frac{1910}{38341}a^{11}-\frac{2480}{38341}a^{10}-\frac{138}{1667}a^{9}+\frac{505}{38341}a^{8}+\frac{18836}{38341}a^{7}-\frac{27022}{38341}a^{6}-\frac{12881}{38341}a^{5}+\frac{20416}{38341}a^{4}+\frac{19220}{38341}a^{3}-\frac{67830}{38341}a^{2}+\frac{21122}{38341}a-\frac{39424}{38341}$, $\frac{4323}{191705}a^{11}-\frac{6931}{191705}a^{10}+\frac{6201}{191705}a^{9}-\frac{76}{38341}a^{8}+\frac{42917}{191705}a^{7}-\frac{104312}{191705}a^{6}+\frac{28188}{38341}a^{5}+\frac{18093}{191705}a^{4}+\frac{100808}{191705}a^{3}-\frac{273678}{191705}a^{2}+\frac{628509}{191705}a-\frac{1339}{1667}$, $\frac{10886}{191705}a^{11}-\frac{11272}{191705}a^{10}+\frac{30787}{191705}a^{9}+\frac{5046}{38341}a^{8}+\frac{95819}{191705}a^{7}-\frac{192059}{191705}a^{6}+\frac{68239}{38341}a^{5}+\frac{176671}{191705}a^{4}+\frac{52351}{191705}a^{3}-\frac{422676}{191705}a^{2}+\frac{1096988}{191705}a-\frac{25085}{38341}$, $\frac{421}{8335}a^{11}-\frac{21571}{191705}a^{10}-\frac{114}{191705}a^{9}+\frac{2241}{38341}a^{8}+\frac{124857}{191705}a^{7}-\frac{256677}{191705}a^{6}+\frac{23294}{38341}a^{5}+\frac{140818}{191705}a^{4}+\frac{172593}{191705}a^{3}-\frac{694528}{191705}a^{2}+\frac{635539}{191705}a-\frac{52327}{38341}$ | 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: | \( 103.136133603 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 103.136133603 \cdot 2}{4\cdot\sqrt{38806720086016}}\cr\approx \mathstrut & 0.509338601920 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T24):
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, 6.0.33856.2, 6.2.778688.4, 6.0.778688.1 |
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: | data not computed |
Degree 16 sibling: | data not computed |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 | $x^{12} + 2 x^{11} + 16 x^{10} + 44 x^{9} + 18 x^{8} - 8 x^{7} + 24 x^{6} + 40 x^{5} + 20 x^{4} + 8 x^{3} + 8$ | $4$ | $3$ | $18$ | $A_4 \times C_2$ | $[2, 2, 2]^{3}$ |
\(23\) | 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |