Normalized defining polynomial
\( x^{15} - x^{14} - x^{13} + 3x^{12} - x^{10} - 4x^{9} + 4x^{8} + 2x^{7} - 7x^{6} - x^{5} + 2x^{4} - x^{3} + 2x + 1 \)
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: | \(12853809661090929\) \(\medspace = 3\cdot 4284603220363643\) | 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: | \(11.86\) | 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: | $3^{1/2}4284603220363643^{1/2}\approx 113374642.9369942$ | ||
Ramified primes: | \(3\), \(4284603220363643\) | 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: | $\Q(\sqrt{12853\!\cdots\!90929}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{709}a^{14}+\frac{128}{709}a^{13}+\frac{204}{709}a^{12}+\frac{86}{709}a^{11}-\frac{250}{709}a^{10}-\frac{346}{709}a^{9}+\frac{29}{709}a^{8}+\frac{200}{709}a^{7}+\frac{278}{709}a^{6}-\frac{304}{709}a^{5}-\frac{222}{709}a^{4}-\frac{276}{709}a^{3}-\frac{155}{709}a^{2}-\frac{143}{709}a-\frac{11}{709}$
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)
| |
Fundamental units: | $a$, $\frac{1453}{709}a^{14}-\frac{2610}{709}a^{13}+\frac{759}{709}a^{12}+\frac{3719}{709}a^{11}-\frac{3078}{709}a^{10}+\frac{1361}{709}a^{9}-\frac{6784}{709}a^{8}+\frac{11254}{709}a^{7}-\frac{6577}{709}a^{6}-\frac{4968}{709}a^{5}+\frac{2865}{709}a^{4}-\frac{443}{709}a^{3}-\frac{1880}{709}a^{2}+\frac{667}{709}a+\frac{1742}{709}$, $\frac{1514}{709}a^{14}-\frac{2601}{709}a^{13}+\frac{441}{709}a^{12}+\frac{4002}{709}a^{11}-\frac{2730}{709}a^{10}+\frac{816}{709}a^{9}-\frac{7142}{709}a^{8}+\frac{11401}{709}a^{7}-\frac{5217}{709}a^{6}-\frac{5787}{709}a^{5}+\frac{2085}{709}a^{4}+\frac{446}{709}a^{3}-\frac{700}{709}a^{2}+\frac{452}{709}a+\frac{1780}{709}$, $\frac{2077}{709}a^{14}-\frac{3564}{709}a^{13}+\frac{435}{709}a^{12}+\frac{5626}{709}a^{11}-\frac{3807}{709}a^{10}+\frac{993}{709}a^{9}-\frac{9958}{709}a^{8}+\frac{14815}{709}a^{7}-\frac{6101}{709}a^{6}-\frac{8906}{709}a^{5}+\frac{3301}{709}a^{4}+\frac{329}{709}a^{3}-\frac{758}{709}a^{2}+\frac{2187}{709}a+\frac{2677}{709}$, $\frac{291}{709}a^{14}-\frac{329}{709}a^{13}-\frac{192}{709}a^{12}+\frac{920}{709}a^{11}-\frac{432}{709}a^{10}-\frac{8}{709}a^{9}-\frac{778}{709}a^{8}+\frac{771}{709}a^{7}+\frac{72}{709}a^{6}-\frac{1966}{709}a^{5}+\frac{1335}{709}a^{4}-\frac{199}{709}a^{3}-\frac{1147}{709}a^{2}+\frac{927}{709}a+\frac{1053}{709}$, $\frac{2833}{709}a^{14}-\frac{4638}{709}a^{13}+\frac{97}{709}a^{12}+\frac{8250}{709}a^{11}-\frac{4922}{709}a^{10}+\frac{329}{709}a^{9}-\frac{12140}{709}a^{8}+\frac{19252}{709}a^{7}-\frac{6506}{709}a^{6}-\frac{14686}{709}a^{5}+\frac{5629}{709}a^{4}+\frac{2246}{709}a^{3}-\frac{3789}{709}a^{2}+\frac{2556}{709}a+\frac{4287}{709}$, $\frac{3490}{709}a^{14}-\frac{5622}{709}a^{13}+\frac{124}{709}a^{12}+\frac{10159}{709}a^{11}-\frac{6102}{709}a^{10}+\frac{596}{709}a^{9}-\frac{14357}{709}a^{8}+\frac{23032}{709}a^{7}-\frac{8200}{709}a^{6}-\frac{18021}{709}a^{5}+\frac{7247}{709}a^{4}+\frac{2418}{709}a^{3}-\frac{5655}{709}a^{2}+\frac{2902}{709}a+\frac{5568}{709}$, $\frac{2786}{709}a^{14}-\frac{4982}{709}a^{13}+\frac{435}{709}a^{12}+\frac{8462}{709}a^{11}-\frac{5934}{709}a^{10}+\frac{284}{709}a^{9}-\frac{12085}{709}a^{8}+\frac{20487}{709}a^{7}-\frac{7519}{709}a^{6}-\frac{15287}{709}a^{5}+\frac{7555}{709}a^{4}+\frac{2456}{709}a^{3}-\frac{2885}{709}a^{2}+\frac{2896}{709}a+\frac{4095}{709}$ | 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: | \( 108.091851993 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 108.091851993 \cdot 1}{2\cdot\sqrt{12853809661090929}}\cr\approx \mathstrut & 0.234647659708 \end{aligned}\]
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 | $15$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | $15$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(4284603220363643\) | $\Q_{4284603220363643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |