Normalized defining polynomial
\( x^{12} + 9x^{10} + 12x^{8} - 35x^{6} - 76x^{4} - 19x^{2} + 19 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2535525376000000\) \(\medspace = 2^{16}\cdot 5^{6}\cdot 19^{5}\) | 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: | \(19.22\) | 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: | $2^{175/96}5^{1/2}19^{5/6}\approx 92.01647650119014$ | ||
Ramified primes: | \(2\), \(5\), \(19\) | 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: | \(\Q(\sqrt{19}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{698}a^{10}+\frac{15}{698}a^{8}+\frac{51}{349}a^{6}-\frac{1}{2}a^{5}-\frac{121}{698}a^{4}-\frac{1}{2}a^{3}+\frac{245}{698}a^{2}-\frac{1}{2}a-\frac{147}{349}$, $\frac{1}{698}a^{11}+\frac{15}{698}a^{9}+\frac{51}{349}a^{7}+\frac{114}{349}a^{5}-\frac{1}{2}a^{4}-\frac{52}{349}a^{3}-\frac{1}{2}a^{2}+\frac{55}{698}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{50}{349}a^{10}-\frac{401}{349}a^{8}-\frac{214}{349}a^{6}+\frac{1862}{349}a^{4}+\frac{1710}{349}a^{2}-\frac{656}{349}$, $\frac{55}{698}a^{10}-\frac{238}{349}a^{8}-\frac{375}{698}a^{6}+\frac{1408}{349}a^{4}+\frac{2579}{698}a^{2}-\frac{989}{349}$, $\frac{181}{698}a^{10}-\frac{1319}{698}a^{8}+\frac{35}{698}a^{6}+\frac{5847}{698}a^{4}+\frac{3119}{698}a^{2}-\frac{1579}{698}$, $\frac{79}{698}a^{10}+\frac{487}{698}a^{8}-\frac{667}{698}a^{6}-\frac{2579}{698}a^{4}+\frac{1207}{698}a^{2}+\frac{1553}{698}$, $\frac{69}{349}a^{10}+\frac{1023}{698}a^{8}+\frac{58}{349}a^{6}-\frac{4483}{698}a^{4}-\frac{1941}{349}a^{2}+\frac{261}{698}$, $\frac{54}{349}a^{11}+\frac{1}{698}a^{10}-\frac{461}{349}a^{9}+\frac{15}{698}a^{8}-\frac{895}{698}a^{7}+\frac{51}{349}a^{6}+\frac{1997}{349}a^{5}+\frac{114}{349}a^{4}+\frac{3173}{349}a^{3}-\frac{401}{349}a^{2}+\frac{691}{698}a-\frac{2039}{698}$, $\frac{43}{349}a^{11}-\frac{87}{698}a^{10}-\frac{296}{349}a^{9}-\frac{607}{698}a^{8}+\frac{151}{349}a^{7}+\frac{100}{349}a^{6}+\frac{3077}{698}a^{5}+\frac{2849}{698}a^{4}+\frac{917}{698}a^{3}+\frac{1719}{698}a^{2}-\frac{1589}{698}a-\frac{124}{349}$ | 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: | \( 1034.9193313912845 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 1034.9193313912845 \cdot 1}{2\cdot\sqrt{2535525376000000}}\cr\approx \mathstrut & 0.256260682804949 \end{aligned}\]
Galois group
$A_4^2:D_4$ (as 12T208):
A solvable group of order 1152 |
The 44 conjugacy class representatives for $A_4^2:D_4$ |
Character table for $A_4^2:D_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 6.6.722000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.16.1 | $x^{12} - 2 x^{11} + 4 x^{10} + 4 x^{9} - 4 x^{7} + 10 x^{6} + 8 x^{5} + 4 x^{4} + 12 x^{3} + 12 x^{2} + 12$ | $6$ | $2$ | $16$ | 12T208 | $[4/3, 4/3, 4/3, 4/3, 2, 2]_{3}^{6}$ |
\(5\) | 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(19\) | 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
19.6.5.3 | $x^{6} + 190$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |