Normalized defining polynomial
\( x^{12} - 6 x^{11} + 25 x^{10} - 70 x^{9} + 172 x^{8} - 334 x^{7} + 551 x^{6} - 712 x^{5} + 777 x^{4} + \cdots + 54 \)
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: | \(1452563570950144\) \(\medspace = 2^{16}\cdot 53^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.34\) | 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: | $2^{3/2}53^{1/2}\approx 20.591260281974$ | ||
Ramified primes: | \(2\), \(53\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{603}a^{10}-\frac{5}{603}a^{9}+\frac{26}{603}a^{8}-\frac{74}{603}a^{7}+\frac{254}{603}a^{6}+\frac{79}{603}a^{5}-\frac{86}{201}a^{4}+\frac{107}{603}a^{3}-\frac{61}{603}a^{2}-\frac{23}{201}a-\frac{1}{67}$, $\frac{1}{181503}a^{11}+\frac{145}{181503}a^{10}-\frac{19618}{181503}a^{9}+\frac{28348}{181503}a^{8}-\frac{5821}{181503}a^{7}-\frac{6290}{25929}a^{6}-\frac{25616}{60501}a^{5}-\frac{63517}{181503}a^{4}-\frac{1942}{4221}a^{3}+\frac{740}{20167}a^{2}+\frac{10282}{60501}a+\frac{2396}{20167}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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)
| |
Fundamental units: | $\frac{4619}{181503}a^{11}-\frac{15622}{181503}a^{10}+\frac{53785}{181503}a^{9}-\frac{77761}{181503}a^{8}+\frac{174808}{181503}a^{7}-\frac{7870}{25929}a^{6}-\frac{19478}{60501}a^{5}+\frac{753682}{181503}a^{4}-\frac{20360}{4221}a^{3}+\frac{453620}{60501}a^{2}-\frac{269821}{60501}a+\frac{55641}{20167}$, $\frac{1663}{181503}a^{11}-\frac{4180}{181503}a^{10}+\frac{620}{60501}a^{9}+\frac{15787}{60501}a^{8}-\frac{59552}{60501}a^{7}+\frac{25469}{8643}a^{6}-\frac{1189196}{181503}a^{5}+\frac{30899}{2709}a^{4}-\frac{60779}{4221}a^{3}+\frac{2504906}{181503}a^{2}-\frac{571841}{60501}a+\frac{75461}{20167}$, $\frac{4568}{181503}a^{11}-\frac{25124}{181503}a^{10}+\frac{10758}{20167}a^{9}-\frac{82423}{60501}a^{8}+\frac{189032}{60501}a^{7}-\frac{47942}{8643}a^{6}+\frac{1455824}{181503}a^{5}-\frac{1573946}{181503}a^{4}+\frac{32636}{4221}a^{3}-\frac{907307}{181503}a^{2}+\frac{101272}{60501}a-\frac{3947}{20167}$, $\frac{2999}{20167}a^{11}-\frac{138367}{181503}a^{10}+\frac{532298}{181503}a^{9}-\frac{1307171}{181503}a^{8}+\frac{3041168}{181503}a^{7}-\frac{739487}{25929}a^{6}+\frac{7572434}{181503}a^{5}-\frac{2539640}{60501}a^{4}+\frac{156964}{4221}a^{3}-\frac{3165767}{181503}a^{2}+\frac{76977}{20167}a+\frac{94197}{20167}$, $\frac{2803}{181503}a^{11}-\frac{4430}{181503}a^{10}+\frac{3446}{181503}a^{9}+\frac{47818}{181503}a^{8}-\frac{129988}{181503}a^{7}+\frac{61996}{25929}a^{6}-\frac{299399}{60501}a^{5}+\frac{1352732}{181503}a^{4}-\frac{35821}{4221}a^{3}+\frac{460015}{60501}a^{2}-\frac{268796}{60501}a+\frac{28069}{20167}$ | 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: | \( 357.3814526 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 357.3814526 \cdot 4}{2\cdot\sqrt{1452563570950144}}\cr\approx \mathstrut & 1.153914528 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{-53}) \), 3.1.212.1 x3, 6.0.38112512.1, 6.2.179776.1, 6.0.9528128.1 |
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: | 4.2.848.1 |
Degree 6 siblings: | 6.2.179776.1, 6.0.38112512.1 |
Degree 8 sibling: | 8.0.32319410176.2 |
Degree 12 sibling: | 12.2.27406859829248.2 |
Minimal sibling: | 4.2.848.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.9 | $x^{4} + 2 x^{3} + 6$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ |
2.4.6.9 | $x^{4} + 2 x^{3} + 6$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(53\) | 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |