Normalized defining polynomial
\( x^{10} - 2x^{9} + 9x^{8} - 24x^{7} + 23x^{6} - 108x^{5} + 5x^{4} - 231x^{3} - 108x^{2} - 220x - 151 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1969153020026821\) \(\medspace = 17^{3}\cdot 73^{3}\cdot 101^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.84\) | 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: | $17^{1/2}73^{1/2}101^{1/2}\approx 354.0353089735542$ | ||
Ramified primes: | \(17\), \(73\), \(101\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{125341}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{12400469}a^{9}-\frac{4234500}{12400469}a^{8}+\frac{2810761}{12400469}a^{7}-\frac{5679767}{12400469}a^{6}+\frac{4367109}{12400469}a^{5}-\frac{4918415}{12400469}a^{4}-\frac{3221240}{12400469}a^{3}+\frac{4444324}{12400469}a^{2}+\frac{3885107}{12400469}a+\frac{1192352}{12400469}$
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)
| |
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: | $\frac{13236}{288383}a^{9}-\frac{29184}{288383}a^{8}+\frac{95298}{288383}a^{7}-\frac{273657}{288383}a^{6}+\frac{142970}{288383}a^{5}-\frac{850903}{288383}a^{4}-\frac{59622}{288383}a^{3}-\frac{1022174}{288383}a^{2}-\frac{891925}{288383}a-\frac{76986}{288383}$, $\frac{393916}{12400469}a^{9}-\frac{614934}{12400469}a^{8}+\frac{3054473}{12400469}a^{7}-\frac{8878716}{12400469}a^{6}+\frac{6646350}{12400469}a^{5}-\frac{42688456}{12400469}a^{4}+\frac{4815523}{12400469}a^{3}-\frac{94683919}{12400469}a^{2}-\frac{10473092}{12400469}a-\frac{92837164}{12400469}$, $\frac{329083}{12400469}a^{9}+\frac{740375}{12400469}a^{8}-\frac{2121485}{12400469}a^{7}+\frac{7928709}{12400469}a^{6}-\frac{35824646}{12400469}a^{5}+\frac{16853249}{12400469}a^{4}-\frac{100434207}{12400469}a^{3}-\frac{34241782}{12400469}a^{2}-\frac{102092178}{12400469}a-\frac{104471103}{12400469}$, $\frac{1136328}{12400469}a^{9}-\frac{2128992}{12400469}a^{8}+\frac{7227154}{12400469}a^{7}-\frac{18575615}{12400469}a^{6}-\frac{1050544}{12400469}a^{5}-\frac{53702289}{12400469}a^{4}-\frac{64369176}{12400469}a^{3}-\frac{92005951}{12400469}a^{2}-\frac{133509098}{12400469}a-\frac{96284174}{12400469}$, $\frac{154050}{12400469}a^{9}+\frac{1946745}{12400469}a^{8}-\frac{1844492}{12400469}a^{7}+\frac{8986290}{12400469}a^{6}-\frac{34304145}{12400469}a^{5}-\frac{25575319}{12400469}a^{4}-\frac{138859186}{12400469}a^{3}-\frac{118186449}{12400469}a^{2}-\frac{144308094}{12400469}a-\frac{130326387}{12400469}$ | 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: | \( 5625.61945556 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 5625.61945556 \cdot 2}{2\cdot\sqrt{1969153020026821}}\cr\approx \mathstrut & 0.790332939523 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_{6}$ |
Character table for $S_{6}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 6 siblings: | 6.2.125341.1, 6.2.1969153020026821.1 |
Degree 12 siblings: | data not computed |
Degree 15 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.2.125341.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
17.4.2.2 | $x^{4} - 272 x^{2} + 867$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(73\) | 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
73.4.0.1 | $x^{4} + 16 x^{2} + 56 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
73.4.2.2 | $x^{4} - 5110 x^{2} + 26645$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(101\) | 101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
101.4.2.2 | $x^{4} - 9797 x^{2} + 20402$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
101.4.0.1 | $x^{4} + x^{2} + 78 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |