Normalized defining polynomial
\( x^{10} - 4x^{9} - 2x^{8} - 10x^{7} + 207x^{6} - 172x^{5} - 481x^{4} - 1904x^{3} + 2339x^{2} - 11530x - 24155 \)
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: | \(174158864690625\) \(\medspace = 3^{5}\cdot 5^{5}\cdot 47^{5}\) | 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: | \(26.55\) | 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: | $3^{1/2}5^{1/2}47^{1/2}\approx 26.551836094703507$ | ||
Ramified primes: | \(3\), \(5\), \(47\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{705}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{92}a^{8}+\frac{5}{92}a^{7}+\frac{11}{92}a^{6}-\frac{1}{46}a^{5}-\frac{1}{46}a^{4}+\frac{3}{23}a^{3}+\frac{13}{92}a^{2}+\frac{7}{46}a+\frac{1}{46}$, $\frac{1}{515640164948948}a^{9}+\frac{861998786055}{515640164948948}a^{8}+\frac{8399887256581}{257820082474474}a^{7}-\frac{38672429007749}{515640164948948}a^{6}-\frac{16305367133569}{515640164948948}a^{5}+\frac{107186536914551}{257820082474474}a^{4}+\frac{162967514557051}{515640164948948}a^{3}-\frac{78422338269333}{257820082474474}a^{2}+\frac{50073707065337}{515640164948948}a+\frac{56142354108161}{257820082474474}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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)
| |
Fundamental units: | $\frac{18844999807}{257820082474474}a^{9}-\frac{244541759837}{515640164948948}a^{8}+\frac{245674547887}{515640164948948}a^{7}-\frac{49967942025}{257820082474474}a^{6}+\frac{3902320057839}{257820082474474}a^{5}-\frac{881244841569}{22419137606476}a^{4}-\frac{9388342303693}{515640164948948}a^{3}-\frac{130628450439}{5604784401619}a^{2}+\frac{106057254981155}{515640164948948}a-\frac{82434056186886}{128910041237237}$, $\frac{31466382137}{515640164948948}a^{9}-\frac{90533929951}{257820082474474}a^{8}+\frac{6784590565}{11209568803238}a^{7}-\frac{1201534707119}{515640164948948}a^{6}+\frac{6351318351777}{515640164948948}a^{5}-\frac{4707204727695}{257820082474474}a^{4}+\frac{9010003158881}{257820082474474}a^{3}-\frac{23473712130672}{128910041237237}a^{2}-\frac{39324522385495}{515640164948948}a-\frac{78795420395223}{515640164948948}$, $\frac{41925397281}{515640164948948}a^{9}+\frac{908493702861}{515640164948948}a^{8}-\frac{788107349119}{128910041237237}a^{7}-\frac{2709020860040}{128910041237237}a^{6}-\frac{2941954935613}{128910041237237}a^{5}+\frac{243771379122719}{515640164948948}a^{4}+\frac{271885668984081}{515640164948948}a^{3}-\frac{202906492178614}{128910041237237}a^{2}-\frac{47\!\cdots\!07}{515640164948948}a-\frac{31\!\cdots\!87}{515640164948948}$, $\frac{40809166964}{128910041237237}a^{9}+\frac{43358578720}{128910041237237}a^{8}-\frac{680112708375}{257820082474474}a^{7}-\frac{13728288836907}{515640164948948}a^{6}+\frac{257000327957}{22419137606476}a^{5}+\frac{42533488910197}{515640164948948}a^{4}+\frac{43108229065833}{257820082474474}a^{3}-\frac{33001724951682}{128910041237237}a^{2}+\frac{476354249905879}{257820082474474}a+\frac{14\!\cdots\!47}{515640164948948}$, $\frac{385690783793}{515640164948948}a^{9}-\frac{103529394121}{128910041237237}a^{8}+\frac{1336343254661}{515640164948948}a^{7}-\frac{3239396589302}{128910041237237}a^{6}+\frac{2208446970581}{22419137606476}a^{5}+\frac{16658416466586}{128910041237237}a^{4}+\frac{462945984806683}{515640164948948}a^{3}+\frac{336821943493831}{515640164948948}a^{2}+\frac{12\!\cdots\!41}{515640164948948}a+\frac{571971603739582}{128910041237237}$ | 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: | \( 795.995187483 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 795.995187483 \cdot 2}{2\cdot\sqrt{174158864690625}}\cr\approx \mathstrut & 0.376025341475 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{705}) \), 5.1.2209.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.30331310150327427376962890625.1 |
Degree 10 sibling: | 10.0.3705507759375.1 |
Minimal sibling: | 10.0.3705507759375.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | R | R | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.10.5.1 | $x^{10} + 162 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(47\) | 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.47.2t1.a.a | $1$ | $ 47 $ | \(\Q(\sqrt{-47}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.705.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 47 $ | \(\Q(\sqrt{705}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 2.10575.10t3.b.b | $2$ | $ 3^{2} \cdot 5^{2} \cdot 47 $ | 10.2.174158864690625.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.47.5t2.a.b | $2$ | $ 47 $ | 5.1.2209.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.47.5t2.a.a | $2$ | $ 47 $ | 5.1.2209.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.10575.10t3.b.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 47 $ | 10.2.174158864690625.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |