Normalized defining polynomial
\( x^{10} - 3x^{9} + 9x^{8} - 27x^{7} - 18x^{6} + 252x^{5} - 558x^{4} + 585x^{3} - 171x^{2} + 161x - 87 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(9227446944279201\) \(\medspace = 3^{16}\cdot 11^{8}\) | 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: | \(39.49\) | 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: | $3^{11/6}11^{4/5}\approx 51.031278439357436$ | ||
Ramified primes: | \(3\), \(11\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11}a^{5}+\frac{4}{11}a^{4}+\frac{2}{11}a^{3}-\frac{5}{11}a^{2}-\frac{2}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{6}-\frac{3}{11}a^{4}-\frac{2}{11}a^{3}-\frac{4}{11}a^{2}-\frac{2}{11}a-\frac{4}{11}$, $\frac{1}{66}a^{7}+\frac{5}{33}a^{4}+\frac{4}{11}a^{3}-\frac{1}{11}a^{2}-\frac{16}{33}a+\frac{1}{22}$, $\frac{1}{132}a^{8}-\frac{1}{132}a^{7}-\frac{1}{22}a^{6}-\frac{1}{66}a^{5}-\frac{4}{33}a^{4}-\frac{7}{22}a^{3}+\frac{29}{66}a^{2}-\frac{61}{132}a+\frac{3}{44}$, $\frac{1}{76296}a^{9}+\frac{19}{19074}a^{8}+\frac{233}{76296}a^{7}+\frac{419}{9537}a^{6}+\frac{611}{38148}a^{5}+\frac{9091}{38148}a^{4}+\frac{31}{102}a^{3}+\frac{8269}{76296}a^{2}+\frac{3316}{9537}a+\frac{15}{25432}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{10}$, which has order $10$
Unit group
Rank: | $5$ | 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: | $\frac{19}{38148}a^{9}-\frac{1}{38148}a^{8}+\frac{23}{9537}a^{7}-\frac{235}{19074}a^{6}-\frac{409}{9537}a^{5}-\frac{61}{1734}a^{4}-\frac{149}{1122}a^{3}+\frac{3941}{38148}a^{2}-\frac{13567}{38148}a+\frac{865}{6358}$, $\frac{1081}{25432}a^{9}-\frac{1963}{19074}a^{8}+\frac{25027}{76296}a^{7}-\frac{3103}{3179}a^{6}-\frac{49619}{38148}a^{5}+\frac{34003}{3468}a^{4}-\frac{6773}{374}a^{3}+\frac{1217903}{76296}a^{2}+\frac{7034}{9537}a+\frac{180429}{25432}$, $\frac{24}{3179}a^{9}-\frac{943}{38148}a^{8}+\frac{2657}{38148}a^{7}-\frac{1233}{6358}a^{6}-\frac{2345}{19074}a^{5}+\frac{19586}{9537}a^{4}-\frac{1729}{374}a^{3}+\frac{67393}{19074}a^{2}+\frac{925}{3468}a+\frac{24261}{12716}$, $\frac{3031}{38148}a^{9}-\frac{6913}{38148}a^{8}+\frac{1943}{3179}a^{7}-\frac{32743}{19074}a^{6}-\frac{23539}{9537}a^{5}+\frac{114579}{6358}a^{4}-\frac{36557}{1122}a^{3}+\frac{1008713}{38148}a^{2}-\frac{3109}{12716}a+\frac{80677}{6358}$, $\frac{451}{6936}a^{9}-\frac{1993}{12716}a^{8}+\frac{37483}{76296}a^{7}-\frac{27691}{19074}a^{6}-\frac{26349}{12716}a^{5}+\frac{585203}{38148}a^{4}-\frac{1415}{51}a^{3}+\frac{563203}{25432}a^{2}+\frac{62429}{38148}a+\frac{280761}{25432}$ | 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: | \( 7960.83309692 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 7960.83309692 \cdot 10}{2\cdot\sqrt{9227446944279201}}\cr\approx \mathstrut & 2.58325458868 \end{aligned}\]
Galois group
A non-solvable group of order 60 |
The 5 conjugacy class representatives for $A_{5}$ |
Character table for $A_{5}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.1.10673289.1 |
Degree 6 sibling: | 6.2.864536409.2 |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Minimal sibling: | 5.1.10673289.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{3}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.3.5.3 | $x^{3} + 9 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
3.6.11.8 | $x^{6} + 18 x^{2} + 21$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ | |
\(11\) | 11.5.4.3 | $x^{5} + 33$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.5.4.3 | $x^{5} + 33$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |