Normalized defining polynomial
\( x^{12} - 8x^{10} - 3x^{9} + 21x^{8} + 14x^{7} - 21x^{6} - 18x^{5} + 5x^{4} + 3x^{3} + x^{2} + 3x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(52649684000000\) \(\medspace = 2^{8}\cdot 5^{6}\cdot 19^{4}\cdot 101\) | 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: | \(13.91\) | 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: | $2^{2/3}5^{1/2}19^{2/3}101^{1/2}\approx 254.00060640965427$ | ||
Ramified primes: | \(2\), \(5\), \(19\), \(101\) | 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: | \(\Q(\sqrt{101}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{37}a^{11}+\frac{3}{37}a^{10}+\frac{1}{37}a^{9}-\frac{16}{37}a^{7}+\frac{3}{37}a^{6}-\frac{12}{37}a^{5}-\frac{17}{37}a^{4}-\frac{9}{37}a^{3}+\frac{13}{37}a^{2}+\frac{3}{37}a+\frac{12}{37}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{2}{37}a^{11}+\frac{6}{37}a^{10}+\frac{2}{37}a^{9}-a^{8}-\frac{106}{37}a^{7}+\frac{6}{37}a^{6}+\frac{272}{37}a^{5}+\frac{151}{37}a^{4}-\frac{166}{37}a^{3}-\frac{122}{37}a^{2}+\frac{6}{37}a-\frac{13}{37}$, $\frac{39}{37}a^{11}-\frac{31}{37}a^{10}-\frac{294}{37}a^{9}+3a^{8}+\frac{782}{37}a^{7}-\frac{31}{37}a^{6}-\frac{912}{37}a^{5}-\frac{71}{37}a^{4}+\frac{389}{37}a^{3}-\frac{85}{37}a^{2}+\frac{6}{37}a+\frac{61}{37}$, $a$, $\frac{34}{37}a^{11}-\frac{9}{37}a^{10}-\frac{262}{37}a^{9}-a^{8}+\frac{677}{37}a^{7}+\frac{287}{37}a^{6}-\frac{704}{37}a^{5}-\frac{319}{37}a^{4}+\frac{212}{37}a^{3}-\frac{76}{37}a^{2}+\frac{28}{37}a+\frac{75}{37}$, $\frac{21}{37}a^{11}-\frac{11}{37}a^{10}-\frac{164}{37}a^{9}+a^{8}+\frac{441}{37}a^{7}-\frac{11}{37}a^{6}-\frac{548}{37}a^{5}-\frac{24}{37}a^{4}+\frac{292}{37}a^{3}-\frac{60}{37}a^{2}+\frac{26}{37}a+\frac{67}{37}$, $\frac{11}{37}a^{11}-\frac{4}{37}a^{10}-\frac{63}{37}a^{9}+\frac{83}{37}a^{7}-\frac{4}{37}a^{6}-\frac{21}{37}a^{5}+\frac{109}{37}a^{4}+\frac{12}{37}a^{3}-\frac{116}{37}a^{2}-\frac{4}{37}a-\frac{16}{37}$, $\frac{10}{37}a^{11}-\frac{7}{37}a^{10}-\frac{101}{37}a^{9}+a^{8}+\frac{358}{37}a^{7}-\frac{7}{37}a^{6}-\frac{527}{37}a^{5}-\frac{133}{37}a^{4}+\frac{280}{37}a^{3}+\frac{56}{37}a^{2}+\frac{67}{37}a+\frac{46}{37}$, $\frac{39}{37}a^{11}-\frac{31}{37}a^{10}-\frac{294}{37}a^{9}+3a^{8}+\frac{782}{37}a^{7}-\frac{31}{37}a^{6}-\frac{912}{37}a^{5}-\frac{71}{37}a^{4}+\frac{389}{37}a^{3}-\frac{85}{37}a^{2}+\frac{6}{37}a+\frac{24}{37}$, $a^{11}-a^{10}-7a^{9}+4a^{8}+17a^{7}-3a^{6}-18a^{5}+5a^{3}-2a^{2}+3a+1$ | 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: | \( 238.420109994 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 238.420109994 \cdot 1}{2\cdot\sqrt{52649684000000}}\cr\approx \mathstrut & 0.166040757456 \end{aligned}\]
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 | ${\href{/padicField/3.12.0.1}{12} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.8.2 | $x^{12} - 8 x^{3} + 16$ | $3$ | $4$ | $8$ | $C_3\times (C_3 : C_4)$ | $[\ ]_{3}^{12}$ |
\(5\) | 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(19\) | 19.6.4.2 | $x^{6} - 342 x^{3} + 722$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(101\) | $\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
101.3.0.1 | $x^{3} + 3 x + 99$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
101.3.0.1 | $x^{3} + 3 x + 99$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |