Normalized defining polynomial
\( x^{10} + 967 x^{8} - 1267 x^{7} + 281810 x^{6} - 211065 x^{5} + 32568404 x^{4} + 6546983 x^{3} + \cdots + 18644013911 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-141538897855111828137102439\) \(\medspace = -\,13^{5}\cdot 23^{4}\cdot 103\cdot 421^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(412.18\) | 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))
| |
Galois root discriminant: | $13^{1/2}23^{2/3}103^{1/2}421^{1/2}\approx 6072.2534985245575$ | ||
Ramified primes: | \(13\), \(23\), \(103\), \(421\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-563719}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | deg 32$^{16}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{8}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{47\!\cdots\!65}a^{9}-\frac{70\!\cdots\!69}{94\!\cdots\!93}a^{8}+\frac{31\!\cdots\!41}{47\!\cdots\!65}a^{7}+\frac{20\!\cdots\!58}{47\!\cdots\!65}a^{6}+\frac{44\!\cdots\!62}{94\!\cdots\!93}a^{5}+\frac{88\!\cdots\!41}{47\!\cdots\!65}a^{4}+\frac{16\!\cdots\!32}{94\!\cdots\!93}a^{3}-\frac{73\!\cdots\!03}{94\!\cdots\!93}a^{2}+\frac{21\!\cdots\!78}{47\!\cdots\!65}a+\frac{12\!\cdots\!71}{47\!\cdots\!65}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{139138}$, which has order $1113104$ (assuming GRH)
Unit group
Rank: | $4$ | 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: | $\frac{27\!\cdots\!83}{47\!\cdots\!65}a^{9}+\frac{17\!\cdots\!93}{47\!\cdots\!65}a^{8}+\frac{51\!\cdots\!63}{94\!\cdots\!93}a^{7}-\frac{19\!\cdots\!23}{47\!\cdots\!65}a^{6}+\frac{68\!\cdots\!24}{47\!\cdots\!65}a^{5}-\frac{16\!\cdots\!19}{47\!\cdots\!65}a^{4}+\frac{66\!\cdots\!56}{47\!\cdots\!65}a^{3}+\frac{44\!\cdots\!96}{47\!\cdots\!65}a^{2}+\frac{14\!\cdots\!47}{47\!\cdots\!65}a-\frac{10\!\cdots\!87}{47\!\cdots\!65}$, $\frac{26\!\cdots\!08}{47\!\cdots\!65}a^{9}-\frac{29\!\cdots\!11}{47\!\cdots\!65}a^{8}+\frac{28\!\cdots\!64}{47\!\cdots\!65}a^{7}-\frac{60\!\cdots\!67}{47\!\cdots\!65}a^{6}+\frac{94\!\cdots\!12}{47\!\cdots\!65}a^{5}-\frac{16\!\cdots\!73}{47\!\cdots\!65}a^{4}+\frac{10\!\cdots\!23}{47\!\cdots\!65}a^{3}+\frac{10\!\cdots\!78}{47\!\cdots\!65}a^{2}+\frac{24\!\cdots\!23}{47\!\cdots\!65}a-\frac{38\!\cdots\!32}{47\!\cdots\!65}$, $\frac{25\!\cdots\!14}{47\!\cdots\!65}a^{9}-\frac{15\!\cdots\!27}{47\!\cdots\!65}a^{8}+\frac{23\!\cdots\!06}{47\!\cdots\!65}a^{7}-\frac{91\!\cdots\!16}{94\!\cdots\!93}a^{6}+\frac{63\!\cdots\!99}{47\!\cdots\!65}a^{5}-\frac{74\!\cdots\!63}{47\!\cdots\!65}a^{4}+\frac{62\!\cdots\!46}{47\!\cdots\!65}a^{3}+\frac{14\!\cdots\!01}{47\!\cdots\!65}a^{2}+\frac{27\!\cdots\!11}{94\!\cdots\!93}a+\frac{14\!\cdots\!59}{47\!\cdots\!65}$, $\frac{40\!\cdots\!98}{47\!\cdots\!65}a^{9}+\frac{26\!\cdots\!67}{47\!\cdots\!65}a^{8}+\frac{58\!\cdots\!91}{47\!\cdots\!65}a^{7}+\frac{12\!\cdots\!41}{47\!\cdots\!65}a^{6}+\frac{24\!\cdots\!86}{47\!\cdots\!65}a^{5}-\frac{63\!\cdots\!76}{94\!\cdots\!93}a^{4}+\frac{31\!\cdots\!44}{47\!\cdots\!65}a^{3}+\frac{24\!\cdots\!04}{47\!\cdots\!65}a^{2}+\frac{73\!\cdots\!31}{47\!\cdots\!65}a+\frac{12\!\cdots\!43}{47\!\cdots\!65}$ (assuming GRH) | 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: | \( 831.0761696414519 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 831.0761696414519 \cdot 1113104}{2\cdot\sqrt{141538897855111828137102439}}\cr\approx \mathstrut & 0.380722035996555 \end{aligned}\] (assuming GRH)
Galois group
$C_2\wr S_5$ (as 10T39):
A non-solvable group of order 3840 |
The 36 conjugacy class representatives for $C_2 \wr S_5$ |
Character table for $C_2 \wr S_5$ |
Intermediate fields
5.5.2895217.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 10.0.863374992140167.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.2.0.1}{2} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(23\) | 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.6.4.1 | $x^{6} + 63 x^{5} + 1338 x^{4} + 9937 x^{3} + 8139 x^{2} + 31314 x + 206412$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(103\) | 103.2.1.1 | $x^{2} + 309$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
103.3.0.1 | $x^{3} + 2 x + 98$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
103.3.0.1 | $x^{3} + 2 x + 98$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(421\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |