Normalized defining polynomial
\( x^{12} - 3 x^{11} - 12 x^{10} + 26 x^{9} + 28 x^{8} + 42 x^{7} + 221 x^{6} - 1012 x^{5} - 830 x^{4} + \cdots + 14641 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(6964478817623209\) \(\medspace = 19^{6}\cdot 23^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(20.90\) | 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))
| |
Galois root discriminant: | $19^{1/2}23^{1/2}\approx 20.904544960366874$ | ||
Ramified primes: | \(19\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11}a^{9}-\frac{3}{11}a^{8}-\frac{1}{11}a^{7}+\frac{4}{11}a^{6}-\frac{5}{11}a^{5}-\frac{2}{11}a^{4}+\frac{1}{11}a^{3}-\frac{5}{11}a$, $\frac{1}{121}a^{10}-\frac{3}{121}a^{9}-\frac{12}{121}a^{8}+\frac{26}{121}a^{7}+\frac{28}{121}a^{6}+\frac{42}{121}a^{5}-\frac{21}{121}a^{4}-\frac{4}{11}a^{3}+\frac{17}{121}a^{2}+\frac{5}{11}a$, $\frac{1}{33\!\cdots\!59}a^{11}+\frac{971997462530454}{33\!\cdots\!59}a^{10}+\frac{13\!\cdots\!91}{33\!\cdots\!59}a^{9}+\frac{42\!\cdots\!71}{33\!\cdots\!59}a^{8}-\frac{16\!\cdots\!40}{33\!\cdots\!59}a^{7}-\frac{14\!\cdots\!16}{33\!\cdots\!59}a^{6}+\frac{16\!\cdots\!38}{33\!\cdots\!59}a^{5}+\frac{10\!\cdots\!89}{30\!\cdots\!69}a^{4}+\frac{25\!\cdots\!52}{33\!\cdots\!59}a^{3}-\frac{620368832046464}{30\!\cdots\!69}a^{2}+\frac{278277704111589}{27\!\cdots\!79}a-\frac{72168759941190}{248049863269189}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}$, which has order $4$
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)
| |
Fundamental units: | $\frac{15401889182910}{33\!\cdots\!59}a^{11}+\frac{15453746136095}{33\!\cdots\!59}a^{10}-\frac{442396945556917}{33\!\cdots\!59}a^{9}+\frac{225844803163250}{33\!\cdots\!59}a^{8}+\frac{804397811995992}{33\!\cdots\!59}a^{7}+\frac{22\!\cdots\!14}{33\!\cdots\!59}a^{6}+\frac{74\!\cdots\!04}{33\!\cdots\!59}a^{5}-\frac{14\!\cdots\!44}{30\!\cdots\!69}a^{4}-\frac{23\!\cdots\!86}{33\!\cdots\!59}a^{3}+\frac{15\!\cdots\!45}{30\!\cdots\!69}a^{2}+\frac{711219880283930}{27\!\cdots\!79}a+\frac{9028591348936}{248049863269189}$, $\frac{32055270389645}{33\!\cdots\!59}a^{11}-\frac{199137785324546}{33\!\cdots\!59}a^{10}-\frac{58024741536121}{33\!\cdots\!59}a^{9}+\frac{14\!\cdots\!46}{33\!\cdots\!59}a^{8}-\frac{412042281170901}{33\!\cdots\!59}a^{7}+\frac{18\!\cdots\!92}{33\!\cdots\!59}a^{6}-\frac{19\!\cdots\!55}{33\!\cdots\!59}a^{5}-\frac{50\!\cdots\!10}{30\!\cdots\!69}a^{4}+\frac{47\!\cdots\!88}{33\!\cdots\!59}a^{3}+\frac{12\!\cdots\!12}{30\!\cdots\!69}a^{2}-\frac{12\!\cdots\!64}{27\!\cdots\!79}a+\frac{63098854710162}{248049863269189}$, $\frac{222839038007954}{33\!\cdots\!59}a^{11}-\frac{10\!\cdots\!50}{33\!\cdots\!59}a^{10}-\frac{12\!\cdots\!71}{33\!\cdots\!59}a^{9}+\frac{87\!\cdots\!22}{33\!\cdots\!59}a^{8}-\frac{10\!\cdots\!03}{33\!\cdots\!59}a^{7}+\frac{22\!\cdots\!32}{33\!\cdots\!59}a^{6}+\frac{27\!\cdots\!68}{33\!\cdots\!59}a^{5}-\frac{23\!\cdots\!04}{27\!\cdots\!79}a^{4}+\frac{30\!\cdots\!94}{33\!\cdots\!59}a^{3}+\frac{35\!\cdots\!01}{30\!\cdots\!69}a^{2}-\frac{11\!\cdots\!96}{27\!\cdots\!79}a+\frac{18\!\cdots\!58}{248049863269189}$, $\frac{97551695560999}{33\!\cdots\!59}a^{11}-\frac{204187225100446}{33\!\cdots\!59}a^{10}-\frac{720342848873844}{33\!\cdots\!59}a^{9}-\frac{516942973847077}{33\!\cdots\!59}a^{8}-\frac{15\!\cdots\!09}{33\!\cdots\!59}a^{7}+\frac{16\!\cdots\!62}{33\!\cdots\!59}a^{6}+\frac{20\!\cdots\!67}{33\!\cdots\!59}a^{5}-\frac{10\!\cdots\!69}{30\!\cdots\!69}a^{4}+\frac{92\!\cdots\!97}{33\!\cdots\!59}a^{3}-\frac{60\!\cdots\!98}{30\!\cdots\!69}a^{2}+\frac{35\!\cdots\!00}{27\!\cdots\!79}a+\frac{665559020469035}{248049863269189}$, $\frac{3391080046478}{32\!\cdots\!59}a^{11}-\frac{17348644554459}{32\!\cdots\!59}a^{10}-\frac{6173104059994}{32\!\cdots\!59}a^{9}+\frac{118457520862957}{32\!\cdots\!59}a^{8}-\frac{179513597556770}{32\!\cdots\!59}a^{7}+\frac{470238512635314}{32\!\cdots\!59}a^{6}-\frac{21927922825586}{32\!\cdots\!59}a^{5}-\frac{399922284813999}{297168648074969}a^{4}+\frac{66\!\cdots\!57}{32\!\cdots\!59}a^{3}+\frac{260660251636299}{297168648074969}a^{2}-\frac{91035181546581}{27015331643179}a+\frac{21830673586283}{2455939240289}$ | 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: | \( 155.18383819 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 155.18383819 \cdot 4}{2\cdot\sqrt{6964478817623209}}\cr\approx \mathstrut & 0.22882916932 \end{aligned}\]
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
\(\Q(\sqrt{437}) \), 3.1.23.1, 6.2.83453453.1, 6.0.10051.1, 6.0.4392287.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.231173.1, 6.0.10051.1 |
Degree 8 siblings: | 8.0.68939809.1, 8.4.36469158961.1 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.10051.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | R | R | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(19\) | 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.8.4.1 | $x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
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 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |