Normalized defining polynomial
\( x^{12} - x^{11} - 2x^{10} + x^{9} + 4x^{8} - 5x^{7} - x^{6} + 12x^{5} - 10x^{4} - 6x^{3} + 16x^{2} - 16x + 8 \)
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: | \(75957575873792\) \(\medspace = 2^{8}\cdot 197^{5}\) | 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: | \(14.35\) | 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: | $2^{2/3}197^{1/2}\approx 22.280235493786417$ | ||
Ramified primes: | \(2\), \(197\) | 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: | \(\Q(\sqrt{197}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{236}a^{11}-\frac{51}{236}a^{10}-\frac{12}{59}a^{9}+\frac{41}{236}a^{8}-\frac{10}{59}a^{7}+\frac{107}{236}a^{6}-\frac{41}{236}a^{5}-\frac{31}{118}a^{4}-\frac{24}{59}a^{3}+\frac{37}{118}a^{2}+\frac{23}{59}a+\frac{26}{59}$
Monogenic: | Not computed | |
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)
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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)
| |
Fundamental units: | $\frac{35}{236}a^{11}-\frac{15}{236}a^{10}-\frac{73}{118}a^{9}-\frac{99}{236}a^{8}+\frac{63}{59}a^{7}+\frac{87}{236}a^{6}-\frac{255}{236}a^{5}+\frac{95}{118}a^{4}+\frac{149}{118}a^{3}-\frac{239}{118}a^{2}-\frac{21}{59}a+\frac{25}{59}$, $\frac{4}{59}a^{11}+\frac{5}{118}a^{10}-\frac{15}{59}a^{9}-\frac{13}{59}a^{8}+\frac{17}{59}a^{7}+\frac{15}{59}a^{6}-\frac{46}{59}a^{5}+\frac{35}{118}a^{4}+\frac{29}{59}a^{3}-\frac{58}{59}a^{2}+\frac{14}{59}a+\frac{3}{59}$, $\frac{7}{236}a^{11}-\frac{3}{236}a^{10}+\frac{9}{118}a^{9}+\frac{51}{236}a^{8}-\frac{11}{59}a^{7}-\frac{195}{236}a^{6}-\frac{51}{236}a^{5}+\frac{39}{59}a^{4}-\frac{41}{118}a^{3}+\frac{23}{118}a^{2}+\frac{102}{59}a+\frac{5}{59}$, $a-1$, $\frac{2}{59}a^{11}-\frac{27}{118}a^{10}-\frac{15}{118}a^{9}+\frac{23}{59}a^{8}+\frac{17}{118}a^{7}-\frac{81}{59}a^{6}+\frac{13}{118}a^{5}+\frac{283}{118}a^{4}-\frac{74}{59}a^{3}-\frac{88}{59}a^{2}+\frac{125}{59}a-\frac{87}{59}$ | 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: | \( 282.237835404 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 282.237835404 \cdot 1}{2\cdot\sqrt{75957575873792}}\cr\approx \mathstrut & 0.996274235205 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
3.3.788.1, 4.0.788.1 x2, 6.2.620944.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.0.788.1 |
Degree 6 siblings: | 6.2.122325968.1, 6.2.620944.1 |
Degree 8 sibling: | 8.0.24098215696.1 |
Degree 12 sibling: | 12.4.14963642447137024.1 |
Minimal sibling: | 4.0.788.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.1.0.1}{1} }^{12}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(197\) | 197.2.0.1 | $x^{2} + 192 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
197.2.1.2 | $x^{2} + 394$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |