Normalized defining polynomial
\( x^{10} - 4x^{9} + 7x^{8} - 18x^{7} + 79x^{6} - 205x^{5} + 297x^{4} - 250x^{3} + 130x^{2} - 45x + 9 \)
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)
| |
Discriminant: | \(-49064203594375\) \(\medspace = -\,5^{4}\cdot 151^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.39\) | 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: | $5^{2/3}151^{1/2}\approx 35.93093151785668$ | ||
Ramified primes: | \(5\), \(151\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-151}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{17}a^{8}-\frac{8}{17}a^{7}-\frac{8}{17}a^{6}-\frac{1}{17}a^{5}-\frac{5}{17}a^{3}-\frac{6}{17}a^{2}-\frac{8}{17}a+\frac{2}{17}$, $\frac{1}{51}a^{9}-\frac{1}{51}a^{8}+\frac{4}{51}a^{7}-\frac{2}{17}a^{6}+\frac{10}{51}a^{5}-\frac{22}{51}a^{4}-\frac{8}{17}a^{3}-\frac{16}{51}a^{2}-\frac{20}{51}a-\frac{1}{17}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{7}$, which has order $7$
Unit group
Rank: | $4$ | 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)
| |
Fundamental units: | $a^{9}-3a^{8}+4a^{7}-14a^{6}+65a^{5}-140a^{4}+157a^{3}-93a^{2}+37a-8$, $\frac{19}{51}a^{9}-\frac{76}{51}a^{8}+\frac{124}{51}a^{7}-\frac{107}{17}a^{6}+\frac{1471}{51}a^{5}-\frac{3784}{51}a^{4}+\frac{1711}{17}a^{3}-\frac{3736}{51}a^{2}+\frac{1453}{51}a-\frac{159}{17}$, $\frac{19}{51}a^{9}-\frac{76}{51}a^{8}+\frac{124}{51}a^{7}-\frac{107}{17}a^{6}+\frac{1471}{51}a^{5}-\frac{3784}{51}a^{4}+\frac{1711}{17}a^{3}-\frac{3736}{51}a^{2}+\frac{1453}{51}a-\frac{142}{17}$, $\frac{19}{51}a^{9}-\frac{67}{51}a^{8}+\frac{103}{51}a^{7}-\frac{97}{17}a^{6}+\frac{80}{3}a^{5}-\frac{3274}{51}a^{4}+\frac{1373}{17}a^{3}-\frac{2719}{51}a^{2}+\frac{1075}{51}a-7$ | 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: | \( 130.469796492 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 130.469796492 \cdot 7}{2\cdot\sqrt{49064203594375}}\cr\approx \mathstrut & 0.638403110262 \end{aligned}\]
Galois group
$C_2\times A_5$ (as 10T11):
A non-solvable group of order 120 |
The 10 conjugacy class representatives for $A_5\times C_2$ |
Character table for $A_5\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-151}) \), 5.1.570025.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 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 | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(151\) | 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
151.4.2.1 | $x^{4} + 298 x^{3} + 22515 x^{2} + 46786 x + 3373376$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
151.4.2.1 | $x^{4} + 298 x^{3} + 22515 x^{2} + 46786 x + 3373376$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.151.2t1.a.a | $1$ | $ 151 $ | \(\Q(\sqrt{-151}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
3.570025.12t33.a.a | $3$ | $ 5^{2} \cdot 151^{2}$ | 5.1.570025.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
3.570025.12t33.a.b | $3$ | $ 5^{2} \cdot 151^{2}$ | 5.1.570025.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
3.3775.12t76.a.a | $3$ | $ 5^{2} \cdot 151 $ | 10.0.49064203594375.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ | |
3.3775.12t76.a.b | $3$ | $ 5^{2} \cdot 151 $ | 10.0.49064203594375.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ | |
* | 4.570025.5t4.a.a | $4$ | $ 5^{2} \cdot 151^{2}$ | 5.1.570025.1 | $A_5$ (as 5T4) | $1$ | $0$ |
* | 4.570025.10t11.a.a | $4$ | $ 5^{2} \cdot 151^{2}$ | 10.0.49064203594375.1 | $A_5\times C_2$ (as 10T11) | $1$ | $0$ |
5.14250625.6t12.a.a | $5$ | $ 5^{4} \cdot 151^{2}$ | 5.1.570025.1 | $A_5$ (as 5T4) | $1$ | $1$ | |
5.2151844375.12t75.a.a | $5$ | $ 5^{4} \cdot 151^{3}$ | 10.0.49064203594375.1 | $A_5\times C_2$ (as 10T11) | $1$ | $-1$ |