Normalized defining polynomial
\( x^{10} - 3x^{9} - 2x^{8} + 16x^{7} + 32x^{6} - 134x^{5} + 138x^{4} + x^{3} + 449x^{2} - 922x + 1576 \)
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: | \(-13759056816473143\) \(\medspace = -\,7^{5}\cdot 904793^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(41.10\) | 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: | $7^{1/2}904793^{1/2}\approx 2516.654724033474$ | ||
Ramified primes: | \(7\), \(904793\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{16}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{69463406604}a^{9}-\frac{329932891}{34731703302}a^{8}+\frac{6994178117}{17365851651}a^{7}-\frac{1007879335}{5788617217}a^{6}+\frac{7793946929}{17365851651}a^{5}-\frac{2300205707}{11577234434}a^{4}-\frac{1660735781}{5788617217}a^{3}+\frac{19205079457}{69463406604}a^{2}-\frac{2958787499}{11577234434}a-\frac{4219834387}{17365851651}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{106}$, which has order $106$
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: | $\frac{71115416}{17365851651}a^{9}-\frac{89103676}{17365851651}a^{8}-\frac{525803780}{17365851651}a^{7}+\frac{127347480}{5788617217}a^{6}+\frac{4168082560}{17365851651}a^{5}-\frac{843020859}{5788617217}a^{4}-\frac{2047367770}{5788617217}a^{3}-\frac{8093121631}{17365851651}a^{2}+\frac{8437406770}{5788617217}a-\frac{10687789841}{17365851651}$, $\frac{104111561}{34731703302}a^{9}+\frac{83058341}{17365851651}a^{8}+\frac{394024264}{17365851651}a^{7}-\frac{55010084}{5788617217}a^{6}-\frac{3136032770}{17365851651}a^{5}+\frac{835128476}{5788617217}a^{4}+\frac{1665547366}{5788617217}a^{3}+\frac{15263667703}{34731703302}a^{2}-\frac{6453188268}{5788617217}a+\frac{61228530559}{17365851651}$, $\frac{38119271}{34731703302}a^{9}-\frac{6045335}{17365851651}a^{8}-\frac{131779516}{17365851651}a^{7}+\frac{72337396}{5788617217}a^{6}+\frac{1032049790}{17365851651}a^{5}-\frac{7892383}{5788617217}a^{4}-\frac{381820404}{5788617217}a^{3}-\frac{922575559}{34731703302}a^{2}+\frac{1984218502}{5788617217}a+\frac{67906592369}{17365851651}$, $\frac{4993382}{17365851651}a^{9}+\frac{42302714}{17365851651}a^{8}-\frac{271670456}{17365851651}a^{7}+\frac{107417727}{5788617217}a^{6}+\frac{91094467}{17365851651}a^{5}+\frac{691470039}{5788617217}a^{4}-\frac{2050524371}{5788617217}a^{3}+\frac{1643045240}{17365851651}a^{2}+\frac{1894048172}{5788617217}a-\frac{10320346997}{17365851651}$ | 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: | \( 870.5650124401076 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 870.5650124401076 \cdot 106}{2\cdot\sqrt{13759056816473143}}\cr\approx \mathstrut & 3.85196283695866 \end{aligned}\]
Galois group
$C_2\times S_5$ (as 10T22):
A non-solvable group of order 240 |
The 14 conjugacy class representatives for $S_5\times C_2$ |
Character table for $S_5\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 5.5.904793.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 30 siblings: | 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 | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{4}$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.10.5.2 | $x^{10} + 35 x^{8} + 492 x^{6} + 8 x^{5} + 3360 x^{4} - 560 x^{3} + 11516 x^{2} + 1968 x + 17516$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(904793\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |