Normalized defining polynomial
\( x^{10} - 4x^{9} + 3x^{8} + 9x^{7} - 15x^{6} - 3x^{5} + 21x^{4} - 9x^{3} - 3x^{2} + 2x + 1 \)
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: | \(-31050641088\) \(\medspace = -\,2^{6}\cdot 3^{9}\cdot 157^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.20\) | 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: | $2^{3/2}3^{9/10}157^{2/3}\approx 221.25200976714402$ | ||
Ramified primes: | \(2\), \(3\), \(157\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{226}a^{9}+\frac{39}{226}a^{8}-\frac{15}{226}a^{7}-\frac{71}{226}a^{6}-\frac{17}{226}a^{5}+\frac{57}{226}a^{4}+\frac{99}{226}a^{3}+\frac{67}{226}a^{2}+\frac{53}{226}a-\frac{46}{113}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{58}{113} a^{9} - \frac{224}{113} a^{8} + \frac{147}{113} a^{7} + \frac{515}{113} a^{6} - \frac{760}{113} a^{5} - \frac{197}{113} a^{4} + \frac{996}{113} a^{3} - \frac{408}{113} a^{2} + \frac{23}{113} a + \frac{88}{113} \) (order $6$) | 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: | $a$, $\frac{71}{226}a^{9}-\frac{141}{113}a^{8}+\frac{89}{113}a^{7}+\frac{361}{113}a^{6}-\frac{547}{113}a^{5}-\frac{180}{113}a^{4}+\frac{859}{113}a^{3}-\frac{390}{113}a^{2}-\frac{209}{113}a+\frac{361}{226}$, $\frac{153}{226}a^{9}-\frac{350}{113}a^{8}+\frac{378}{113}a^{7}+\frac{614}{113}a^{6}-\frac{1470}{113}a^{5}+\frac{236}{113}a^{4}+\frac{1641}{113}a^{3}-\frac{1372}{113}a^{2}+\frac{156}{113}a+\frac{275}{226}$, $\frac{191}{226}a^{9}-\frac{687}{226}a^{8}+\frac{299}{226}a^{7}+\frac{1807}{226}a^{6}-\frac{2117}{226}a^{5}-\frac{1317}{226}a^{4}+\frac{3315}{226}a^{3}-\frac{537}{226}a^{2}-\frac{499}{226}a+\frac{141}{113}$ | 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: | \( 50.1360011729 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 50.1360011729 \cdot 1}{6\cdot\sqrt{31050641088}}\cr\approx \mathstrut & 0.464368227888 \end{aligned}\]
Galois group
$\PGOPlus(4,5)$ (as 10T41):
A non-solvable group of order 14400 |
The 25 conjugacy class representatives for $(A_5^2 : C_2):C_2$ |
Character table for $(A_5^2 : C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 25 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | 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 | R | R | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(3\) | 3.10.9.2 | $x^{10} + 3$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $[\ ]_{10}^{4}$ |
\(157\) | $\Q_{157}$ | $x + 152$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
157.2.0.1 | $x^{2} + 152 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
157.3.2.1 | $x^{3} + 157$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
157.4.0.1 | $x^{4} + 11 x^{2} + 136 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |