Normalized defining polynomial
\( x^{15} - 2x^{14} - 3x^{13} + 9x^{12} - x^{11} - 5x^{10} + 4x^{9} - x^{8} + x^{7} - x^{5} - x^{4} + 7x^{3} - x^{2} + 1 \)
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: | \(146928604515779584\) \(\medspace = 2^{10}\cdot 3461^{4}\) | 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: | \(13.95\) | 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: | $2^{2/3}3461^{1/2}\approx 93.38722346966051$ | ||
Ramified primes: | \(2\), \(3461\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{47524}a^{14}+\frac{2413}{47524}a^{13}+\frac{2851}{23762}a^{12}-\frac{11621}{47524}a^{11}-\frac{897}{23762}a^{10}-\frac{7831}{47524}a^{9}-\frac{21071}{47524}a^{8}-\frac{3006}{11881}a^{7}-\frac{795}{47524}a^{6}+\frac{4797}{47524}a^{5}+\frac{3165}{11881}a^{4}+\frac{15967}{47524}a^{3}-\frac{2707}{23762}a^{2}+\frac{18051}{47524}a-\frac{10105}{47524}$
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: | $a$, $\frac{2319}{11881}a^{14}-\frac{204}{11881}a^{13}-\frac{61865}{47524}a^{12}+\frac{23679}{47524}a^{11}+\frac{146709}{47524}a^{10}-\frac{5921}{11881}a^{9}-\frac{71551}{47524}a^{8}-\frac{7677}{47524}a^{7}+\frac{9831}{11881}a^{6}+\frac{26389}{47524}a^{5}-\frac{9525}{47524}a^{4}-\frac{5604}{11881}a^{3}+\frac{24485}{47524}a^{2}+\frac{120953}{47524}a+\frac{18991}{47524}$, $\frac{3566}{11881}a^{14}-\frac{8967}{11881}a^{13}-\frac{6940}{11881}a^{12}+\frac{36085}{11881}a^{11}-\frac{17307}{11881}a^{10}-\frac{21873}{23762}a^{9}+\frac{8139}{11881}a^{8}-\frac{10936}{11881}a^{7}+\frac{44821}{23762}a^{6}-\frac{2538}{11881}a^{5}-\frac{14121}{11881}a^{4}-\frac{26503}{23762}a^{3}+\frac{35944}{11881}a^{2}-\frac{13273}{11881}a+\frac{13167}{23762}$, $\frac{2262}{11881}a^{14}-\frac{7054}{11881}a^{13}-\frac{7487}{47524}a^{12}+\frac{106971}{47524}a^{11}-\frac{85833}{47524}a^{10}-\frac{10183}{23762}a^{9}+\frac{27761}{47524}a^{8}-\frac{46359}{47524}a^{7}+\frac{27125}{23762}a^{6}-\frac{21799}{47524}a^{5}+\frac{26721}{47524}a^{4}-\frac{13653}{23762}a^{3}+\frac{70777}{47524}a^{2}-\frac{73945}{47524}a-\frac{5745}{47524}$, $\frac{6459}{47524}a^{14}-\frac{26067}{47524}a^{13}+\frac{9999}{47524}a^{12}+\frac{43643}{23762}a^{11}-\frac{122281}{47524}a^{10}+\frac{8869}{47524}a^{9}+\frac{23395}{23762}a^{8}-\frac{20681}{47524}a^{7}+\frac{21449}{47524}a^{6}-\frac{30615}{23762}a^{5}+\frac{17779}{47524}a^{4}-\frac{19989}{47524}a^{3}+\frac{44281}{47524}a^{2}-\frac{22973}{11881}a+\frac{4475}{11881}$, $\frac{4779}{11881}a^{14}-\frac{21329}{23762}a^{13}-\frac{22193}{23762}a^{12}+\frac{42559}{11881}a^{11}-\frac{26531}{23762}a^{10}-\frac{11080}{11881}a^{9}+\frac{5047}{11881}a^{8}-\frac{479}{23762}a^{7}+\frac{14496}{11881}a^{6}-\frac{17348}{11881}a^{5}+\frac{20057}{23762}a^{4}-\frac{17251}{11881}a^{3}+\frac{27074}{11881}a^{2}-\frac{2212}{11881}a-\frac{2941}{23762}$, $\frac{2263}{11881}a^{14}-\frac{30445}{47524}a^{13}-\frac{8441}{47524}a^{12}+\frac{131773}{47524}a^{11}-\frac{52445}{23762}a^{10}-\frac{87333}{47524}a^{9}+\frac{109811}{47524}a^{8}-\frac{17525}{23762}a^{7}+\frac{15427}{47524}a^{6}-\frac{26373}{47524}a^{5}-\frac{2903}{23762}a^{4}+\frac{48443}{47524}a^{3}+\frac{49121}{47524}a^{2}-\frac{73027}{47524}a-\frac{5261}{23762}$, $\frac{1879}{23762}a^{14}-\frac{20911}{47524}a^{13}+\frac{6673}{47524}a^{12}+\frac{86005}{47524}a^{11}-\frac{22123}{11881}a^{10}-\frac{70947}{47524}a^{9}+\frac{49571}{47524}a^{8}+\frac{2283}{11881}a^{7}-\frac{5479}{47524}a^{6}-\frac{67637}{47524}a^{5}-\frac{10692}{11881}a^{4}-\frac{30707}{47524}a^{3}+\frac{77627}{47524}a^{2}-\frac{40495}{47524}a-\frac{1457}{23762}$ | 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: | \( 470.017972215 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 470.017972215 \cdot 1}{2\cdot\sqrt{146928604515779584}}\cr\approx \mathstrut & 0.301786973049 \end{aligned}\]
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: | 6.2.165830644724.1, 6.2.55376.1 |
Degree 10 sibling: | 10.2.2653290315584.1 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 15 sibling: | deg 15 |
Degree 20 siblings: | 20.4.7039949498771842313261056.1, deg 20, 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: | 6.2.55376.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.5.0.1}{5} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
\(3461\) | $\Q_{3461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $2$ | $4$ | $4$ |