Normalized defining polynomial
\( x^{10} - 2x^{9} + 11x^{8} + 79x^{7} + 140x^{6} + 247x^{5} + 481x^{4} - 945x^{3} + 618x^{2} + 612x + 9 \)
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: | \(140848064844913917\) \(\medspace = 3^{4}\cdot 1117^{5}\) | 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: | \(51.87\) | 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: | $3^{1/2}1117^{1/2}\approx 57.88782255362521$ | ||
Ramified primes: | \(3\), \(1117\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{1117}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{1}{9}a^{5}+\frac{1}{9}a^{4}-\frac{4}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{717337058553}a^{9}-\frac{11576992349}{717337058553}a^{8}+\frac{93560195873}{717337058553}a^{7}-\frac{21152371562}{717337058553}a^{6}+\frac{19135790150}{717337058553}a^{5}-\frac{65023111040}{717337058553}a^{4}-\frac{140886947324}{717337058553}a^{3}+\frac{68261197826}{239112352851}a^{2}-\frac{93436823813}{239112352851}a+\frac{3380425691}{79704117617}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{5640824456}{239112352851}a^{9}-\frac{5104463135}{239112352851}a^{8}+\frac{53130419527}{239112352851}a^{7}+\frac{168923132357}{79704117617}a^{6}+\frac{1311486654937}{239112352851}a^{5}+\frac{846600297385}{79704117617}a^{4}+\frac{4712171802824}{239112352851}a^{3}-\frac{1875624282193}{239112352851}a^{2}-\frac{641452867501}{79704117617}a+\frac{390401346289}{79704117617}$, $\frac{4687754209}{717337058553}a^{9}-\frac{5384732998}{239112352851}a^{8}+\frac{23010560953}{239112352851}a^{7}+\frac{94541113853}{239112352851}a^{6}+\frac{22034873883}{79704117617}a^{5}+\frac{163933314860}{239112352851}a^{4}+\frac{827071091561}{717337058553}a^{3}-\frac{500110033121}{79704117617}a^{2}+\frac{6420863835638}{239112352851}a+\frac{33160862438}{79704117617}$, $\frac{1399897252}{239112352851}a^{9}-\frac{2848004236}{239112352851}a^{8}+\frac{14023636675}{239112352851}a^{7}+\frac{109923644006}{239112352851}a^{6}+\frac{177650988115}{239112352851}a^{5}+\frac{195778731860}{239112352851}a^{4}+\frac{62076593347}{79704117617}a^{3}-\frac{808469704709}{79704117617}a^{2}-\frac{427005041579}{79704117617}a-\frac{6149146687}{79704117617}$, $\frac{209278063679}{239112352851}a^{9}-\frac{1569088462937}{717337058553}a^{8}+\frac{7686214803485}{717337058553}a^{7}+\frac{45758916272584}{717337058553}a^{6}+\frac{65062604280788}{717337058553}a^{5}+\frac{122255162974558}{717337058553}a^{4}+\frac{239804904440168}{717337058553}a^{3}-\frac{237673673821298}{239112352851}a^{2}+\frac{248640873003629}{239112352851}a+\frac{1088545883123}{79704117617}$, $\frac{35337015707}{239112352851}a^{9}-\frac{84662620975}{239112352851}a^{8}+\frac{144616300701}{79704117617}a^{7}+\frac{2536256146735}{239112352851}a^{6}+\frac{4417062508079}{239112352851}a^{5}+\frac{6093968571410}{239112352851}a^{4}+\frac{17109244208543}{239112352851}a^{3}-\frac{43164452508200}{239112352851}a^{2}+\frac{13824160175575}{79704117617}a+\frac{1274564559909}{79704117617}$ (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: | \( 288950.6047086043 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 288950.6047086043 \cdot 1}{2\cdot\sqrt{140848064844913917}}\cr\approx \mathstrut & 2.39992480523558 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{1117}) \), 5.1.11229201.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.178543596335013680634836128262546001.1 |
Degree 10 sibling: | 10.0.378284865295203.1 |
Minimal sibling: | 10.0.378284865295203.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }$ | R | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{10}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 | $[\ ]$ |
$\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(1117\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3351.2t1.a.a | $1$ | $ 3 \cdot 1117 $ | \(\Q(\sqrt{-3351}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.1117.2t1.a.a | $1$ | $ 1117 $ | \(\Q(\sqrt{1117}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 2.3351.10t3.a.a | $2$ | $ 3 \cdot 1117 $ | 10.2.140848064844913917.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.3351.5t2.a.a | $2$ | $ 3 \cdot 1117 $ | 5.1.11229201.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.3351.5t2.a.b | $2$ | $ 3 \cdot 1117 $ | 5.1.11229201.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.3351.10t3.a.b | $2$ | $ 3 \cdot 1117 $ | 10.2.140848064844913917.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |