Normalized defining polynomial
\( x^{12} - 6 x^{11} + 19 x^{10} - 40 x^{9} + 42 x^{8} + 6 x^{7} - 108 x^{6} + 201 x^{5} - 192 x^{4} + \cdots + 8 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(288136807515649\) \(\medspace = 257^{6}\) | 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: | \(16.03\) | 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: | $257^{1/2}\approx 16.0312195418814$ | ||
Ramified primes: | \(257\) | 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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{18}a^{8}-\frac{2}{9}a^{7}-\frac{1}{3}a^{6}-\frac{2}{9}a^{5}-\frac{1}{9}a^{4}-\frac{1}{6}a+\frac{2}{9}$, $\frac{1}{18}a^{9}-\frac{2}{9}a^{7}+\frac{4}{9}a^{6}-\frac{4}{9}a^{4}-\frac{1}{6}a^{2}-\frac{4}{9}a-\frac{1}{9}$, $\frac{1}{36}a^{10}-\frac{1}{36}a^{9}+\frac{7}{18}a^{7}+\frac{1}{9}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{5}{12}a^{3}-\frac{5}{36}a^{2}+\frac{1}{3}a$, $\frac{1}{4068}a^{11}+\frac{17}{1356}a^{10}-\frac{1}{339}a^{9}-\frac{23}{2034}a^{8}-\frac{193}{1017}a^{7}-\frac{109}{226}a^{6}+\frac{109}{226}a^{5}+\frac{391}{4068}a^{4}-\frac{53}{4068}a^{3}+\frac{457}{1017}a^{2}-\frac{1}{9}a+\frac{109}{1017}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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)
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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: | $\frac{95}{4068}a^{11}-\frac{51}{226}a^{10}+\frac{363}{452}a^{9}-\frac{203}{113}a^{8}+\frac{4801}{2034}a^{7}+\frac{41}{226}a^{6}-\frac{4874}{1017}a^{5}+\frac{39631}{4068}a^{4}-\frac{9649}{1017}a^{3}+\frac{15347}{4068}a^{2}-\frac{13}{18}a-\frac{1058}{1017}$, $\frac{95}{2034}a^{11}-\frac{1045}{4068}a^{10}+\frac{2579}{4068}a^{9}-\frac{314}{339}a^{8}-\frac{229}{2034}a^{7}+\frac{2968}{1017}a^{6}-\frac{8987}{2034}a^{5}+\frac{974}{339}a^{4}+\frac{7897}{4068}a^{3}-\frac{5551}{1356}a^{2}+\frac{10}{9}a+\frac{16}{113}$, $\frac{95}{4068}a^{11}-\frac{127}{4068}a^{10}-\frac{172}{1017}a^{9}+\frac{295}{339}a^{8}-\frac{2515}{1017}a^{7}+\frac{5567}{2034}a^{6}+\frac{761}{2034}a^{5}-\frac{27943}{4068}a^{4}+\frac{46493}{4068}a^{3}-\frac{8000}{1017}a^{2}+\frac{11}{6}a+\frac{2219}{1017}$, $\frac{605}{2034}a^{11}-\frac{6655}{4068}a^{10}+\frac{19493}{4068}a^{9}-\frac{6301}{678}a^{8}+\frac{14885}{2034}a^{7}+\frac{6412}{1017}a^{6}-\frac{59267}{2034}a^{5}+\frac{14773}{339}a^{4}-\frac{129557}{4068}a^{3}+\frac{15035}{1356}a^{2}+\frac{77}{18}a-\frac{973}{339}$, $\frac{731}{4068}a^{11}-\frac{453}{452}a^{10}+\frac{5671}{2034}a^{9}-\frac{5186}{1017}a^{8}+\frac{3218}{1017}a^{7}+\frac{12191}{2034}a^{6}-\frac{11597}{678}a^{5}+\frac{92365}{4068}a^{4}-\frac{48913}{4068}a^{3}+\frac{1999}{2034}a^{2}+\frac{19}{18}a-\frac{2585}{1017}$, $\frac{631}{2034}a^{11}-\frac{2849}{2034}a^{10}+\frac{6779}{2034}a^{9}-\frac{10381}{2034}a^{8}-\frac{616}{1017}a^{7}+\frac{11303}{1017}a^{6}-\frac{6668}{339}a^{5}+\frac{33151}{2034}a^{4}+\frac{4525}{2034}a^{3}-\frac{8737}{2034}a^{2}+\frac{17}{18}a+\frac{1280}{1017}$, $\frac{169}{4068}a^{11}-\frac{1099}{4068}a^{10}+\frac{1585}{2034}a^{9}-\frac{1435}{1017}a^{8}+\frac{944}{1017}a^{7}+\frac{4163}{2034}a^{6}-\frac{3497}{678}a^{5}+\frac{24947}{4068}a^{4}-\frac{8279}{4068}a^{3}-\frac{529}{678}a^{2}+\frac{17}{18}a+\frac{341}{1017}$ | 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: | \( 447.403572328 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 447.403572328 \cdot 1}{2\cdot\sqrt{288136807515649}}\cr\approx \mathstrut & 0.328631763904 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{257}) \), 3.3.257.1 x3, 6.2.16974593.1, 6.2.66049.1, 6.6.16974593.1 |
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: | 4.0.257.1 |
Degree 6 siblings: | 6.2.66049.1, 6.2.16974593.1 |
Degree 8 sibling: | 8.0.4362470401.1 |
Degree 12 sibling: | 12.0.1121154893057.1 |
Minimal sibling: | 4.0.257.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(257\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |