Normalized defining polynomial
\( x^{18} - 5 x^{17} + 16 x^{16} - 39 x^{15} + 80 x^{14} - 141 x^{13} + 221 x^{12} - 312 x^{11} + 400 x^{10} + \cdots + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-1186012071116620699\) \(\medspace = -\,7^{4}\cdot 139\cdot 1373^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.10\) | 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: | $7^{2/3}139^{1/2}1373^{2/3}\approx 5329.499816940617$ | ||
Ramified primes: | \(7\), \(139\), \(1373\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-139}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $6a^{17}-36a^{16}+112a^{15}-278a^{14}+562a^{13}-992a^{12}+1529a^{11}-2150a^{10}+2713a^{9}-3121a^{8}+3217a^{7}-2984a^{6}+2475a^{5}-1829a^{4}+1123a^{3}-523a^{2}+155a-22$, $6a^{17}-20a^{16}+53a^{15}-107a^{14}+190a^{13}-281a^{12}+382a^{11}-458a^{10}+490a^{9}-453a^{8}+352a^{7}-216a^{6}+96a^{5}+12a^{4}-90a^{3}+96a^{2}-49a+10$, $2a^{17}-a^{16}-2a^{15}+21a^{14}-58a^{13}+137a^{12}-241a^{11}+390a^{10}-545a^{9}+697a^{8}-789a^{7}+801a^{6}-716a^{5}+584a^{4}-408a^{3}+224a^{2}-80a+14$, $11a^{17}-57a^{16}+175a^{15}-421a^{14}+843a^{13}-1461a^{12}+2237a^{11}-3106a^{10}+3892a^{9}-4425a^{8}+4522a^{7}-4146a^{6}+3410a^{5}-2482a^{4}+1494a^{3}-672a^{2}+192a-25$, $13a^{17}-46a^{16}+131a^{15}-282a^{14}+533a^{13}-852a^{12}+1248a^{11}-1630a^{10}+1939a^{9}-2070a^{8}+1980a^{7}-1690a^{6}+1299a^{5}-841a^{4}+416a^{3}-134a^{2}+20a+1$, $7a^{17}-38a^{16}+117a^{15}-283a^{14}+567a^{13}-985a^{12}+1506a^{11}-2092a^{10}+2618a^{9}-2974a^{8}+3031a^{7}-2772a^{6}+2267a^{5}-1645a^{4}+980a^{3}-429a^{2}+114a-13$, $10a^{17}-40a^{16}+114a^{15}-254a^{14}+484a^{13}-792a^{12}+1166a^{11}-1551a^{10}+1862a^{9}-2021a^{8}+1960a^{7}-1701a^{6}+1326a^{5}-889a^{4}+460a^{3}-161a^{2}+30a-2$, $4a^{17}-11a^{16}+33a^{15}-67a^{14}+129a^{13}-201a^{12}+303a^{11}-391a^{10}+477a^{9}-514a^{8}+509a^{7}-452a^{6}+368a^{5}-252a^{4}+151a^{3}-74a^{2}+27a-5$ | 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: | \( 22.1660529321 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 22.1660529321 \cdot 1}{2\cdot\sqrt{1186012071116620699}}\cr\approx \mathstrut & 0.155321877266 \end{aligned}\]
Galois group
$C_2^9.A_9$ (as 18T966):
A non-solvable group of order 92897280 |
The 168 conjugacy class representatives for $C_2^9.A_9$ are not computed |
Character table for $C_2^9.A_9$ is not computed |
Intermediate fields
9.1.92371321.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 36 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 | $18$ | $18$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.5.0.1 | $x^{5} + x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
7.5.0.1 | $x^{5} + x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(139\) | $\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
139.2.1.2 | $x^{2} + 139$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
139.7.0.1 | $x^{7} + 9 x + 137$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
139.7.0.1 | $x^{7} + 9 x + 137$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(1373\) | $\Q_{1373}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1373}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |