Normalized defining polynomial
\( x^{15} - 6 x^{13} - x^{12} + 16 x^{11} + 8 x^{10} - 20 x^{9} - 24 x^{8} + 4 x^{7} + 31 x^{6} + 15 x^{5} + \cdots + 1 \)
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: | \(3276367268581097\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.82\) | 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: | $3276367268581097^{1/2}\approx 57239560.34580539$ | ||
Ramified primes: | \(3276367268581097\) | 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: | $\Q(\sqrt{32763\!\cdots\!81097}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{12}-\frac{2}{7}a^{11}+\frac{3}{7}a^{9}-\frac{3}{7}a^{8}+\frac{2}{7}a^{6}-\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{1}{7}a^{3}+\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{14}-\frac{3}{7}a^{12}+\frac{2}{7}a^{11}+\frac{3}{7}a^{10}+\frac{1}{7}a^{9}+\frac{3}{7}a^{8}+\frac{2}{7}a^{7}-\frac{3}{7}a^{6}+\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}-\frac{3}{7}a^{2}-\frac{3}{7}a-\frac{2}{7}$
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: | $a$, $\frac{3}{7}a^{14}-\frac{3}{7}a^{13}-\frac{26}{7}a^{12}+\frac{12}{7}a^{11}+\frac{86}{7}a^{10}-\frac{6}{7}a^{9}-\frac{150}{7}a^{8}-\frac{78}{7}a^{7}+\frac{104}{7}a^{6}+\frac{184}{7}a^{5}+\frac{33}{7}a^{4}-\frac{131}{7}a^{3}-\frac{54}{7}a^{2}+\frac{25}{7}a+\frac{16}{7}$, $\frac{36}{7}a^{14}-\frac{6}{7}a^{13}-\frac{205}{7}a^{12}+a^{11}+\frac{528}{7}a^{10}+\frac{151}{7}a^{9}-93a^{8}-\frac{607}{7}a^{7}+\frac{216}{7}a^{6}+\frac{869}{7}a^{5}+\frac{244}{7}a^{4}-\frac{489}{7}a^{3}-\frac{254}{7}a^{2}+\frac{107}{7}a+9$, $\frac{13}{7}a^{14}+\frac{4}{7}a^{13}-11a^{12}-\frac{31}{7}a^{11}+\frac{200}{7}a^{10}+\frac{137}{7}a^{9}-\frac{225}{7}a^{8}-\frac{317}{7}a^{7}-\frac{10}{7}a^{6}+\frac{344}{7}a^{5}+\frac{216}{7}a^{4}-\frac{139}{7}a^{3}-20a^{2}+\frac{9}{7}a+\frac{24}{7}$, $\frac{45}{7}a^{14}-\frac{23}{7}a^{13}-\frac{263}{7}a^{12}+\frac{87}{7}a^{11}+\frac{702}{7}a^{10}+\frac{18}{7}a^{9}-\frac{972}{7}a^{8}-\frac{645}{7}a^{7}+\frac{561}{7}a^{6}+\frac{1226}{7}a^{5}+\frac{97}{7}a^{4}-\frac{818}{7}a^{3}-\frac{312}{7}a^{2}+\frac{212}{7}a+\frac{109}{7}$, $6a^{14}-\frac{15}{7}a^{13}-\frac{239}{7}a^{12}+\frac{51}{7}a^{11}+88a^{10}+\frac{74}{7}a^{9}-\frac{788}{7}a^{8}-86a^{7}+\frac{355}{7}a^{6}+\frac{995}{7}a^{5}+\frac{146}{7}a^{4}-\frac{594}{7}a^{3}-\frac{225}{7}a^{2}+\frac{142}{7}a+\frac{68}{7}$, $\frac{45}{7}a^{14}-\frac{22}{7}a^{13}-\frac{255}{7}a^{12}+\frac{78}{7}a^{11}+\frac{660}{7}a^{10}+6a^{9}-\frac{870}{7}a^{8}-\frac{645}{7}a^{7}+\frac{444}{7}a^{6}+160a^{5}+19a^{4}-95a^{3}-\frac{283}{7}a^{2}+23a+\frac{90}{7}$, $\frac{59}{7}a^{14}-4a^{13}-\frac{331}{7}a^{12}+\frac{104}{7}a^{11}+\frac{849}{7}a^{10}+\frac{38}{7}a^{9}-\frac{1104}{7}a^{8}-\frac{792}{7}a^{7}+\frac{565}{7}a^{6}+\frac{1406}{7}a^{5}+\frac{127}{7}a^{4}-\frac{855}{7}a^{3}-\frac{324}{7}a^{2}+\frac{201}{7}a+\frac{99}{7}$ | 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: | \( 37.7909873347 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 37.7909873347 \cdot 1}{2\cdot\sqrt{3276367268581097}}\cr\approx \mathstrut & 0.162491688168 \end{aligned}\]
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 | $15$ | ${\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | $15$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $15$ | $15$ | $15$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3276367268581097\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |