Normalized defining polynomial
\( x^{10} - 2x^{9} + 7x^{8} - 16x^{7} + 24x^{6} - 34x^{5} + 33x^{4} - 16x^{3} + 11x^{2} - 2x + 1 \)
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: | \(-34857992547\) \(\medspace = -\,3^{5}\cdot 7^{2}\cdot 29^{2}\cdot 59^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.33\) | 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: | $3^{1/2}7^{2/3}29^{1/2}59^{1/2}\approx 262.1708037295433$ | ||
Ramified primes: | \(3\), \(7\), \(29\), \(59\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{14}a^{9}-\frac{3}{14}a^{8}+\frac{3}{14}a^{7}+\frac{1}{7}a^{6}+\frac{1}{14}a^{5}+\frac{5}{14}a^{3}-\frac{1}{2}a^{2}+\frac{2}{7}a-\frac{3}{7}$
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: | \( 2 a^{9} - \frac{9}{2} a^{8} + \frac{29}{2} a^{7} - \frac{69}{2} a^{6} + \frac{105}{2} a^{5} - 72 a^{4} + 71 a^{3} - 32 a^{2} + 14 a - \frac{5}{2} \) (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: | $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{5}{2}a^{7}-\frac{9}{2}a^{6}+4a^{5}-5a^{4}+8a^{2}-\frac{3}{2}a+2$, $\frac{29}{14}a^{9}-\frac{31}{14}a^{8}+\frac{143}{14}a^{7}-\frac{271}{14}a^{6}+\frac{239}{14}a^{5}-\frac{43}{2}a^{4}+\frac{13}{7}a^{3}+\frac{61}{2}a^{2}-\frac{19}{7}a+\frac{85}{14}$, $\frac{18}{7}a^{9}-\frac{33}{7}a^{8}+\frac{117}{7}a^{7}-\frac{523}{14}a^{6}+\frac{729}{14}a^{5}-71a^{4}+\frac{859}{14}a^{3}-\frac{31}{2}a^{2}+\frac{151}{14}a+\frac{11}{7}$, $\frac{8}{7}a^{9}-\frac{55}{14}a^{8}+\frac{153}{14}a^{7}-\frac{201}{7}a^{6}+\frac{709}{14}a^{5}-\frac{141}{2}a^{4}+\frac{1123}{14}a^{3}-54a^{2}+\frac{130}{7}a-\frac{55}{7}$ | 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: | \( 52.3378509231 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 52.3378509231 \cdot 1}{6\cdot\sqrt{34857992547}}\cr\approx \mathstrut & 0.457522852144 \end{aligned}\]
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 | ${\href{/padicField/2.10.0.1}{10} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | R | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | R |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.5.0.1 | $x^{5} + x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(29\) | 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
29.4.2.2 | $x^{4} - 696 x^{2} + 1682$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(59\) | 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
59.4.2.2 | $x^{4} - 13924 x^{3} - 56674102 x^{2} - 11222744 x + 6962$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |