Normalized defining polynomial
\( x^{15} + x^{13} - 3x^{12} - 3x^{11} - 5x^{10} - 2x^{9} + 3x^{8} + 7x^{7} + 4x^{6} + x^{5} + x^{4} + 3x^{3} + x^{2} - 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(69184806150882304\) \(\medspace = 2^{10}\cdot 47^{4}\cdot 61^{4}\) | 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: | \(13.26\) | 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: | $2^{2/3}47^{1/2}61^{1/2}\approx 84.99639580651171$ | ||
Ramified primes: | \(2\), \(47\), \(61\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{92}a^{14}+\frac{2}{23}a^{13}-\frac{1}{23}a^{12}+\frac{11}{92}a^{11}-\frac{7}{92}a^{10}+\frac{2}{23}a^{9}+\frac{39}{92}a^{8}-\frac{7}{92}a^{7}+\frac{5}{23}a^{6}-\frac{43}{92}a^{5}-\frac{21}{92}a^{4}+\frac{10}{23}a^{3}-\frac{11}{46}a^{2}-\frac{37}{92}a+\frac{13}{46}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $a$, $\frac{5}{46}a^{14}+\frac{11}{92}a^{13}+\frac{29}{92}a^{12}-\frac{51}{92}a^{11}-\frac{6}{23}a^{10}-\frac{173}{92}a^{9}-\frac{47}{92}a^{8}-\frac{6}{23}a^{7}+\frac{131}{92}a^{6}+\frac{237}{92}a^{5}+\frac{33}{46}a^{4}-\frac{37}{92}a^{3}+\frac{79}{92}a^{2}+\frac{21}{92}a+\frac{19}{23}$, $\frac{4}{23}a^{14}-\frac{5}{46}a^{13}+\frac{5}{92}a^{12}-\frac{77}{92}a^{11}+\frac{3}{92}a^{10}-\frac{14}{23}a^{9}+\frac{141}{92}a^{8}+\frac{95}{92}a^{7}+\frac{34}{23}a^{6}-\frac{67}{92}a^{5}-\frac{221}{92}a^{4}-\frac{25}{46}a^{3}+\frac{85}{92}a^{2}-\frac{63}{92}a-\frac{21}{92}$, $\frac{9}{92}a^{14}+\frac{13}{46}a^{13}+\frac{5}{46}a^{12}+\frac{7}{92}a^{11}-\frac{109}{92}a^{10}-\frac{28}{23}a^{9}-\frac{201}{92}a^{8}-\frac{17}{92}a^{7}+\frac{22}{23}a^{6}+\frac{257}{92}a^{5}+\frac{133}{92}a^{4}+\frac{21}{23}a^{3}+\frac{39}{46}a^{2}+\frac{127}{92}a+\frac{1}{23}$, $\frac{1}{23}a^{14}+\frac{9}{92}a^{13}+\frac{15}{46}a^{12}-\frac{1}{46}a^{11}-\frac{5}{92}a^{10}-\frac{129}{92}a^{9}-\frac{30}{23}a^{8}-\frac{189}{92}a^{7}+\frac{11}{92}a^{6}+\frac{26}{23}a^{5}+\frac{261}{92}a^{4}+\frac{137}{92}a^{3}+\frac{24}{23}a^{2}+\frac{41}{46}a+\frac{35}{92}$, $\frac{43}{92}a^{14}-\frac{1}{92}a^{13}+\frac{29}{46}a^{12}-\frac{125}{92}a^{11}-\frac{35}{23}a^{10}-\frac{231}{92}a^{9}-\frac{163}{92}a^{8}+\frac{34}{23}a^{7}+\frac{285}{92}a^{6}+\frac{267}{92}a^{5}+\frac{33}{23}a^{4}-\frac{5}{92}a^{3}-\frac{13}{46}a^{2}+\frac{19}{92}a+\frac{83}{92}$, $\frac{4}{23}a^{14}-\frac{33}{92}a^{13}+\frac{51}{92}a^{12}-\frac{77}{92}a^{11}+\frac{18}{23}a^{10}-\frac{79}{92}a^{9}+\frac{3}{92}a^{8}-\frac{5}{23}a^{7}-\frac{71}{92}a^{6}-\frac{21}{92}a^{5}+\frac{85}{46}a^{4}+\frac{65}{92}a^{3}+\frac{39}{92}a^{2}+\frac{29}{92}a+\frac{12}{23}$, $\frac{4}{23}a^{14}-\frac{5}{46}a^{13}+\frac{5}{92}a^{12}-\frac{77}{92}a^{11}+\frac{3}{92}a^{10}-\frac{14}{23}a^{9}+\frac{141}{92}a^{8}+\frac{95}{92}a^{7}+\frac{34}{23}a^{6}-\frac{67}{92}a^{5}-\frac{221}{92}a^{4}-\frac{25}{46}a^{3}-\frac{7}{92}a^{2}+\frac{29}{92}a-\frac{21}{92}$ | 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: | \( 273.665310463 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 273.665310463 \cdot 1}{2\cdot\sqrt{69184806150882304}}\cr\approx \mathstrut & 0.256066984661 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_6$ |
Character table for $S_6$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 6 siblings: | 6.0.45872.1, 6.4.94263393452.1 |
Degree 10 sibling: | 10.4.1508214295232.1 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 15 sibling: | deg 15 |
Degree 20 siblings: | deg 20, 20.0.2274710360342158457933824.1, deg 20 |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.0.45872.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.5.0.1}{5} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | ${\href{/padicField/31.5.0.1}{5} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }^{3}$ | R | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
47.4.0.1 | $x^{4} + 8 x^{2} + 40 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
47.8.4.1 | $x^{8} + 204 x^{6} + 80 x^{5} + 14080 x^{4} - 6880 x^{3} + 384824 x^{2} - 499680 x + 3453444$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(61\) | $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
61.6.3.1 | $x^{6} + 7137 x^{5} + 16979120 x^{4} + 13465223383 x^{3} + 1154999059 x^{2} + 95278410610 x + 796785014929$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |