Normalized defining polynomial
\( x^{10} - x^{9} - 3x^{8} + 3x^{7} + 2x^{5} - 3x^{4} - 2x^{3} + 6x^{2} - x - 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(15632169373\) \(\medspace = 63727\cdot 245299\) | 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: | \(10.46\) | 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: | $63727^{1/2}245299^{1/2}\approx 125028.67420316029$ | ||
Ramified primes: | \(63727\), \(245299\) | 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: | $\Q(\sqrt{15632169373}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{113}a^{9}-\frac{42}{113}a^{8}+\frac{24}{113}a^{7}+\frac{36}{113}a^{6}-\frac{7}{113}a^{5}-\frac{50}{113}a^{4}+\frac{13}{113}a^{3}+\frac{30}{113}a^{2}+\frac{19}{113}a+\frac{11}{113}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{79}{113}a^{9}-\frac{41}{113}a^{8}-\frac{251}{113}a^{7}+\frac{132}{113}a^{6}+\frac{12}{113}a^{5}+\frac{118}{113}a^{4}-\frac{103}{113}a^{3}-\frac{229}{113}a^{2}+\frac{371}{113}a+\frac{78}{113}$, $a$, $\frac{24}{113}a^{9}+\frac{9}{113}a^{8}-\frac{102}{113}a^{7}-\frac{40}{113}a^{6}+\frac{58}{113}a^{5}+\frac{43}{113}a^{4}+\frac{86}{113}a^{3}-\frac{71}{113}a^{2}+\frac{4}{113}a+\frac{151}{113}$, $\frac{38}{113}a^{9}-\frac{14}{113}a^{8}-\frac{105}{113}a^{7}+\frac{12}{113}a^{6}-\frac{40}{113}a^{5}+\frac{134}{113}a^{4}-\frac{71}{113}a^{3}+\frac{10}{113}a^{2}+\frac{157}{113}a-\frac{34}{113}$, $\frac{79}{113}a^{9}-\frac{41}{113}a^{8}-\frac{251}{113}a^{7}+\frac{132}{113}a^{6}+\frac{12}{113}a^{5}+\frac{118}{113}a^{4}-\frac{103}{113}a^{3}-\frac{229}{113}a^{2}+\frac{484}{113}a+\frac{78}{113}$, $\frac{53}{113}a^{9}+\frac{34}{113}a^{8}-\frac{197}{113}a^{7}-\frac{126}{113}a^{6}+\frac{81}{113}a^{5}+\frac{175}{113}a^{4}+\frac{124}{113}a^{3}-\frac{218}{113}a^{2}-\frac{10}{113}a+\frac{131}{113}$, $\frac{142}{113}a^{9}-\frac{88}{113}a^{8}-\frac{434}{113}a^{7}+\frac{253}{113}a^{6}+\frac{23}{113}a^{5}+\frac{358}{113}a^{4}-\frac{301}{113}a^{3}-\frac{373}{113}a^{2}+\frac{664}{113}a-\frac{20}{113}$ | 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: | \( 16.520324934712065 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{2}\cdot 16.520324934712065 \cdot 1}{2\cdot\sqrt{15632169373}}\cr\approx \mathstrut & 0.166923958111556 \end{aligned}\]
Galois group
A non-solvable group of order 3628800 |
The 42 conjugacy class representatives for $S_{10}$ |
Character table for $S_{10}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 20 sibling: | data not computed |
Degree 45 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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(63727\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(245299\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |