Normalized defining polynomial
\( x^{15} - 2 x^{14} + 6 x^{13} - 14 x^{12} + 23 x^{11} - 35 x^{10} + 44 x^{9} - 41 x^{8} + 25 x^{7} + \cdots + 1 \)
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: | \(16680932154277329\) \(\medspace = 3\cdot 47\cdot 103\cdot 11057\cdot 103878739\) | 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: | \(12.06\) | 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: | $3^{1/2}47^{1/2}103^{1/2}11057^{1/2}103878739^{1/2}\approx 129154683.05205712$ | ||
Ramified primes: | \(3\), \(47\), \(103\), \(11057\), \(103878739\) | 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: | $\Q(\sqrt{16680\!\cdots\!77329}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{229}a^{14}-\frac{80}{229}a^{13}+\frac{63}{229}a^{12}+\frac{110}{229}a^{11}-\frac{84}{229}a^{10}+\frac{105}{229}a^{9}+\frac{98}{229}a^{8}+\frac{101}{229}a^{7}-\frac{67}{229}a^{6}-\frac{45}{229}a^{5}+\frac{55}{229}a^{4}+\frac{95}{229}a^{3}-\frac{112}{229}a^{2}+\frac{51}{229}a-\frac{91}{229}$
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: | $a$, $\frac{837}{229}a^{14}-\frac{1237}{229}a^{13}+\frac{4412}{229}a^{12}-\frac{9377}{229}a^{11}+\frac{14422}{229}a^{10}-\frac{21806}{229}a^{9}+\frac{25234}{229}a^{8}-\frac{21032}{229}a^{7}+\frac{9644}{229}a^{6}+\frac{2181}{229}a^{5}-\frac{16024}{229}a^{4}+\frac{19975}{229}a^{3}-\frac{14739}{229}a^{2}+\frac{6505}{229}a-\frac{1284}{229}$, $\frac{93}{229}a^{14}-\frac{341}{229}a^{13}+\frac{592}{229}a^{12}-\frac{1907}{229}a^{11}+\frac{2951}{229}a^{10}-\frac{4204}{229}a^{9}+\frac{5679}{229}a^{8}-\frac{4805}{229}a^{7}+\frac{2700}{229}a^{6}+\frac{166}{229}a^{5}-\frac{2671}{229}a^{4}+\frac{4713}{229}a^{3}-\frac{3546}{229}a^{2}+\frac{1995}{229}a-\frac{448}{229}$, $\frac{223}{229}a^{14}+\frac{22}{229}a^{13}+\frac{767}{229}a^{12}-\frac{889}{229}a^{11}+\frac{504}{229}a^{10}-\frac{1317}{229}a^{9}-\frac{130}{229}a^{8}+\frac{1684}{229}a^{7}-\frac{2117}{229}a^{6}+\frac{1644}{229}a^{5}-\frac{2162}{229}a^{4}-\frac{570}{229}a^{3}+\frac{1817}{229}a^{2}-\frac{993}{229}a+\frac{546}{229}$, $\frac{592}{229}a^{14}-\frac{644}{229}a^{13}+\frac{2946}{229}a^{12}-\frac{5641}{229}a^{11}+\frac{8438}{229}a^{10}-\frac{13181}{229}a^{9}+\frac{14277}{229}a^{8}-\frac{11656}{229}a^{7}+\frac{4991}{229}a^{6}+\frac{1527}{229}a^{5}-\frac{10034}{229}a^{4}+\frac{11127}{229}a^{3}-\frac{8138}{229}a^{2}+\frac{3857}{229}a-\frac{744}{229}$, $\frac{52}{229}a^{14}-\frac{38}{229}a^{13}+\frac{299}{229}a^{12}-\frac{463}{229}a^{11}+\frac{899}{229}a^{10}-\frac{1410}{229}a^{9}+\frac{1661}{229}a^{8}-\frac{1847}{229}a^{7}+\frac{1096}{229}a^{6}-\frac{279}{229}a^{5}-\frac{1033}{229}a^{4}+\frac{1505}{229}a^{3}-\frac{1931}{229}a^{2}+\frac{820}{229}a-\frac{152}{229}$, $\frac{769}{229}a^{14}-\frac{1293}{229}a^{13}+\frac{4021}{229}a^{12}-\frac{9300}{229}a^{11}+\frac{13951}{229}a^{10}-\frac{20931}{229}a^{9}+\frac{25211}{229}a^{8}-\frac{20572}{229}a^{7}+\frac{9849}{229}a^{6}+\frac{1577}{229}a^{5}-\frac{14726}{229}a^{4}+\frac{20156}{229}a^{3}-\frac{13993}{229}a^{2}+\frac{6472}{229}a-\frac{1508}{229}$, $\frac{1431}{229}a^{14}-\frac{1812}{229}a^{13}+\frac{7255}{229}a^{12}-\frac{14798}{229}a^{11}+\frac{22005}{229}a^{10}-\frac{34090}{229}a^{9}+\frac{38104}{229}a^{8}-\frac{30425}{229}a^{7}+\frac{13127}{229}a^{6}+\frac{4534}{229}a^{5}-\frac{26177}{229}a^{4}+\frac{30147}{229}a^{3}-\frac{20353}{229}a^{2}+\frac{8632}{229}a-\frac{1523}{229}$ | 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: | \( 124.641970008 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 124.641970008 \cdot 1}{2\cdot\sqrt{16680932154277329}}\cr\approx \mathstrut & 0.237516261054 \end{aligned}\]
Galois group
A non-solvable group of order 1307674368000 |
The 176 conjugacy class representatives for $S_{15}$ |
Character table for $S_{15}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 30 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $15$ | R | $15$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $15$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
3.9.0.1 | $x^{9} + 2 x^{3} + 2 x^{2} + x + 1$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(47\) | 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
47.13.0.1 | $x^{13} + 5 x + 42$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
\(103\) | 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
103.5.0.1 | $x^{5} + 11 x + 98$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
103.8.0.1 | $x^{8} + x^{4} + 70 x^{3} + 71 x^{2} + 49 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
\(11057\) | $\Q_{11057}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(103878739\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |