Normalized defining polynomial
\( x^{12} - 2x^{11} + 7x^{10} - 7x^{9} + 14x^{8} - 6x^{7} + 13x^{6} - 2x^{5} + 10x^{4} - 2x^{3} + 5x^{2} - x + 1 \)
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: | \(44533203125\) \(\medspace = 5^{9}\cdot 151^{2}\) | 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: | \(7.72\) | 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: | $5^{3/4}151^{2/3}\approx 94.81538251069664$ | ||
Ramified primes: | \(5\), \(151\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$
Monogenic: | Yes | |
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)
| |
Torsion generator: | \( -a^{10} + a^{9} - 5 a^{8} + a^{7} - 8 a^{6} - 3 a^{5} - 8 a^{4} - 3 a^{3} - 6 a^{2} - 1 \) (order $10$) | 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: | $a^{8}-a^{7}+4a^{6}+4a^{4}+3a^{3}+4a^{2}+2$, $a^{10}-2a^{9}+6a^{8}-5a^{7}+9a^{6}-2a^{5}+8a^{4}-a^{3}+7a^{2}+2$, $2a^{11}-4a^{10}+14a^{9}-14a^{8}+27a^{7}-11a^{6}+22a^{5}-3a^{4}+15a^{3}-5a^{2}+6a-2$, $2a^{11}-5a^{10}+15a^{9}-18a^{8}+27a^{7}-15a^{6}+18a^{5}-7a^{4}+13a^{3}-8a^{2}+5a-2$, $a^{11}-2a^{10}+7a^{9}-6a^{8}+12a^{7}-a^{6}+9a^{5}+3a^{4}+8a^{3}+a^{2}+2a$ | 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: | \( 6.10223070288 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 6.10223070288 \cdot 1}{10\cdot\sqrt{44533203125}}\cr\approx \mathstrut & 0.177920449060 \end{aligned}\]
Galois group
$C_3\wr C_4$ (as 12T131):
A solvable group of order 324 |
The 36 conjugacy class representatives for $C_3\wr C_4$ |
Character table for $C_3\wr C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 18 siblings: | 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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
\(151\) | $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
151.3.2.3 | $x^{3} + 604$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
151.3.0.1 | $x^{3} + x + 145$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |