Normalized defining polynomial
\( x^{12} - 3 x^{11} + 13 x^{10} - 23 x^{9} + 36 x^{8} - 26 x^{7} + 19 x^{6} + 8 x^{5} + 16 x^{4} - 4 x^{3} + 34 x^{2} + 5 x + 1 \)
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: | \(1998099208210609\) \(\medspace = 7^{6}\cdot 19^{8}\) | 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: | \(18.84\) | 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: | $7^{1/2}19^{2/3}\approx 18.838721275076466$ | ||
Ramified primes: | \(7\), \(19\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{241252}a^{11}+\frac{22743}{241252}a^{10}+\frac{3845}{120626}a^{9}+\frac{9017}{241252}a^{8}-\frac{23795}{241252}a^{7}-\frac{4777}{21932}a^{6}+\frac{48991}{241252}a^{5}-\frac{28580}{60313}a^{4}-\frac{27162}{60313}a^{3}-\frac{102863}{241252}a^{2}-\frac{14967}{60313}a+\frac{12426}{60313}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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)
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Fundamental units: | $\frac{30671}{241252}a^{11}-\frac{44959}{120626}a^{10}+\frac{199019}{120626}a^{9}-\frac{349227}{120626}a^{8}+\frac{558401}{120626}a^{7}-\frac{75403}{21932}a^{6}+\frac{568009}{241252}a^{5}+\frac{204863}{120626}a^{4}+\frac{372633}{241252}a^{3}-\frac{179295}{241252}a^{2}+\frac{1224117}{241252}a-\frac{60317}{241252}$, $\frac{25825}{120626}a^{11}-\frac{160977}{241252}a^{10}+\frac{345419}{120626}a^{9}-\frac{1275509}{241252}a^{8}+\frac{2036893}{241252}a^{7}-\frac{75287}{10966}a^{6}+\frac{670217}{120626}a^{5}+\frac{67663}{120626}a^{4}+\frac{877641}{241252}a^{3}-\frac{131829}{120626}a^{2}+\frac{1573135}{241252}a+\frac{230007}{241252}$, $\frac{2095}{241252}a^{11}-\frac{685}{241252}a^{10}+\frac{6979}{241252}a^{9}+\frac{6323}{120626}a^{8}-\frac{31987}{241252}a^{7}-\frac{335}{5483}a^{6}+\frac{82179}{120626}a^{5}-\frac{118103}{241252}a^{4}-\frac{56611}{241252}a^{3}+\frac{181303}{241252}a^{2}-\frac{53418}{60313}a-\frac{30671}{241252}$, $\frac{4505}{21932}a^{11}-\frac{3643}{5483}a^{10}+\frac{62169}{21932}a^{9}-\frac{117105}{21932}a^{8}+\frac{47020}{5483}a^{7}-\frac{74617}{10966}a^{6}+\frac{26729}{5483}a^{5}+\frac{23039}{21932}a^{4}+\frac{21233}{5483}a^{3}-\frac{40451}{21932}a^{2}+\frac{162437}{21932}a+\frac{11849}{10966}$, $\frac{82321}{241252}a^{11}-\frac{250895}{241252}a^{10}+\frac{272219}{60313}a^{9}-\frac{1973963}{241252}a^{8}+\frac{3153695}{241252}a^{7}-\frac{225977}{21932}a^{6}+\frac{1908443}{241252}a^{5}+\frac{136263}{60313}a^{4}+\frac{625137}{120626}a^{3}-\frac{442953}{241252}a^{2}+\frac{699313}{60313}a-\frac{35781}{120626}$ | 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: | \( 186.316655752 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 186.316655752 \cdot 4}{2\cdot\sqrt{1998099208210609}}\cr\approx \mathstrut & 0.512923222708 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T7):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4 \times C_2$ |
Character table for $A_4 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 3.3.361.1, 6.0.44700103.1, 6.4.912247.1, 6.2.6385729.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.4.912247.1 |
Degree 8 sibling: | 8.0.312900721.1 |
Degree 12 sibling: | 12.4.40777534861441.1 |
Minimal sibling: | 6.4.912247.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(19\) | 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |