Normalized defining polynomial
\( x^{12} - x^{11} - 2 x^{10} + 8 x^{9} - 5 x^{8} - 14 x^{7} + 33 x^{6} + 24 x^{5} - 25 x^{4} + 12 x^{3} + \cdots - 19 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(58824500000000\) \(\medspace = 2^{8}\cdot 5^{9}\cdot 7^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.04\) | 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: | $2^{2/3}5^{3/4}7^{1/2}\approx 14.043106421634382$ | ||
Ramified primes: | \(2\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19}a^{10}+\frac{6}{19}a^{9}+\frac{4}{19}a^{8}-\frac{9}{19}a^{7}-\frac{3}{19}a^{6}+\frac{4}{19}a^{5}-\frac{2}{19}a^{4}-\frac{1}{19}a^{3}+\frac{2}{19}a^{2}+\frac{5}{19}a$, $\frac{1}{1512229}a^{11}-\frac{28241}{1512229}a^{10}+\frac{581155}{1512229}a^{9}+\frac{404145}{1512229}a^{8}-\frac{13818}{79591}a^{7}-\frac{272721}{1512229}a^{6}-\frac{141224}{1512229}a^{5}+\frac{417911}{1512229}a^{4}-\frac{371549}{1512229}a^{3}+\frac{698970}{1512229}a^{2}+\frac{212355}{1512229}a+\frac{31315}{79591}$
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)
| |
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: | $\frac{30295}{1512229}a^{11}-\frac{37436}{1512229}a^{10}-\frac{154794}{1512229}a^{9}+\frac{487200}{1512229}a^{8}-\frac{17918}{79591}a^{7}-\frac{1094032}{1512229}a^{6}+\frac{2659628}{1512229}a^{5}-\frac{483762}{1512229}a^{4}-\frac{3183011}{1512229}a^{3}+\frac{1782011}{1512229}a^{2}-\frac{204987}{1512229}a-\frac{116386}{79591}$, $\frac{8975}{1512229}a^{11}+\frac{34360}{1512229}a^{10}-\frac{130060}{1512229}a^{9}+\frac{159914}{1512229}a^{8}+\frac{598}{4189}a^{7}-\frac{725271}{1512229}a^{6}+\frac{558412}{1512229}a^{5}+\frac{1537579}{1512229}a^{4}-\frac{1142422}{1512229}a^{3}-\frac{584416}{1512229}a^{2}+\frac{2228587}{1512229}a-\frac{63287}{79591}$, $\frac{30295}{1512229}a^{11}-\frac{37436}{1512229}a^{10}-\frac{154794}{1512229}a^{9}+\frac{487200}{1512229}a^{8}-\frac{17918}{79591}a^{7}-\frac{1094032}{1512229}a^{6}+\frac{2659628}{1512229}a^{5}-\frac{483762}{1512229}a^{4}-\frac{3183011}{1512229}a^{3}+\frac{1782011}{1512229}a^{2}-\frac{204987}{1512229}a-\frac{195977}{79591}$, $\frac{254757}{1512229}a^{11}-\frac{441538}{1512229}a^{10}-\frac{122863}{1512229}a^{9}+\frac{2078713}{1512229}a^{8}-\frac{148502}{79591}a^{7}-\frac{1167259}{1512229}a^{6}+\frac{9108900}{1512229}a^{5}-\frac{660752}{1512229}a^{4}-\frac{4773029}{1512229}a^{3}+\frac{8027319}{1512229}a^{2}-\frac{194239}{1512229}a-\frac{327203}{79591}$, $\frac{238641}{1512229}a^{11}-\frac{570102}{1512229}a^{10}+\frac{252037}{1512229}a^{9}+\frac{1729832}{1512229}a^{8}-\frac{195122}{79591}a^{7}+\frac{1217905}{1512229}a^{6}+\frac{7339646}{1512229}a^{5}-\frac{4623967}{1512229}a^{4}-\frac{2212136}{1512229}a^{3}+\frac{7861438}{1512229}a^{2}-\frac{3755605}{1512229}a-\frac{313212}{79591}$, $\frac{48245}{1512229}a^{11}+\frac{31284}{1512229}a^{10}-\frac{414914}{1512229}a^{9}+\frac{807028}{1512229}a^{8}+\frac{4806}{79591}a^{7}-\frac{2544574}{1512229}a^{6}+\frac{3776452}{1512229}a^{5}+\frac{2591396}{1512229}a^{4}-\frac{5467855}{1512229}a^{3}+\frac{2125408}{1512229}a^{2}+\frac{2739958}{1512229}a-\frac{163369}{79591}$, $\frac{296059}{1512229}a^{11}-\frac{365624}{1512229}a^{10}-\frac{551606}{1512229}a^{9}+\frac{2808120}{1512229}a^{8}-\frac{137611}{79591}a^{7}-\frac{3679820}{1512229}a^{6}+\frac{12683282}{1512229}a^{5}+\frac{1227748}{1512229}a^{4}-\frac{7971530}{1512229}a^{3}+\frac{11061152}{1512229}a^{2}+\frac{837227}{1512229}a-\frac{468005}{79591}$ | 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: | \( 155.35343805 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 155.35343805 \cdot 1}{2\cdot\sqrt{58824500000000}}\cr\approx \mathstrut & 0.25255222968 \end{aligned}\]
Galois group
$C_4\times S_3$ (as 12T11):
A solvable group of order 24 |
The 12 conjugacy class representatives for $S_3 \times C_4$ |
Character table for $S_3 \times C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 3.1.140.1, 4.4.6125.1, 6.2.98000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 12 sibling: | 12.0.1200500000000.1 |
Minimal sibling: | 12.0.1200500000000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ |
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(7\) | 7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |