Normalized defining polynomial
\( x^{18} - x^{16} - 4 x^{15} - x^{14} + 3 x^{13} + 9 x^{12} + x^{11} - 2 x^{10} - 13 x^{9} - 2 x^{8} + x^{7} + 9 x^{6} + 3 x^{5} - x^{4} - 4 x^{3} - x^{2} + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(10992670662263790889\) \(\medspace = 23^{6}\cdot 379^{2}\cdot 719^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.42\) | 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: | $23^{1/2}379^{1/2}719^{1/2}\approx 2503.5021469932876$ | ||
Ramified primes: | \(23\), \(379\), \(719\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7}a^{16}+\frac{1}{7}a^{15}-\frac{1}{7}a^{14}+\frac{1}{7}a^{13}+\frac{1}{7}a^{12}+\frac{3}{7}a^{11}-\frac{3}{7}a^{10}+\frac{2}{7}a^{9}+\frac{3}{7}a^{8}+\frac{2}{7}a^{7}-\frac{3}{7}a^{6}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{1}{7}a^{3}-\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{35}a^{17}-\frac{1}{35}a^{16}-\frac{2}{7}a^{15}-\frac{4}{35}a^{14}+\frac{13}{35}a^{13}+\frac{3}{7}a^{12}-\frac{16}{35}a^{11}-\frac{13}{35}a^{10}+\frac{6}{35}a^{9}-\frac{4}{35}a^{8}+\frac{1}{5}a^{7}+\frac{9}{35}a^{6}-\frac{1}{7}a^{5}+\frac{13}{35}a^{4}+\frac{11}{35}a^{3}+\frac{2}{7}a^{2}-\frac{1}{35}a-\frac{9}{35}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{3}{7}a^{17}+\frac{10}{7}a^{16}-\frac{3}{7}a^{15}-\frac{25}{7}a^{14}-\frac{39}{7}a^{13}+\frac{2}{7}a^{12}+\frac{61}{7}a^{11}+\frac{83}{7}a^{10}-\frac{5}{7}a^{9}-\frac{64}{7}a^{8}-\frac{107}{7}a^{7}-\frac{19}{7}a^{6}+\frac{45}{7}a^{5}+\frac{66}{7}a^{4}+\frac{18}{7}a^{3}-\frac{11}{7}a^{2}-\frac{25}{7}a$, $a$, $\frac{23}{35}a^{17}+\frac{27}{35}a^{16}-\frac{1}{7}a^{15}-\frac{107}{35}a^{14}-\frac{141}{35}a^{13}-\frac{5}{7}a^{12}+\frac{202}{35}a^{11}+\frac{251}{35}a^{10}+\frac{19}{5}a^{9}-\frac{187}{35}a^{8}-\frac{299}{35}a^{7}-\frac{188}{35}a^{6}+\frac{209}{35}a^{4}+\frac{93}{35}a^{3}+\frac{1}{7}a^{2}-\frac{43}{35}a-\frac{52}{35}$, $a^{16}-a^{14}-3a^{13}-a^{12}+2a^{11}+6a^{10}-7a^{7}-2a^{6}+a^{5}+2a^{4}+a^{3}-2a$, $\frac{5}{7}a^{16}+\frac{12}{7}a^{15}-\frac{5}{7}a^{14}-\frac{30}{7}a^{13}-\frac{44}{7}a^{12}+\frac{1}{7}a^{11}+\frac{62}{7}a^{10}+\frac{80}{7}a^{9}+\frac{1}{7}a^{8}-\frac{46}{7}a^{7}-\frac{99}{7}a^{6}-\frac{20}{7}a^{5}+\frac{19}{7}a^{4}+\frac{40}{7}a^{3}+\frac{16}{7}a^{2}-\frac{9}{7}a-\frac{9}{7}$, $\frac{1}{35}a^{17}+\frac{34}{35}a^{16}-\frac{9}{7}a^{15}-\frac{39}{35}a^{14}-\frac{57}{35}a^{13}+\frac{24}{7}a^{12}+\frac{124}{35}a^{11}+\frac{92}{35}a^{10}-\frac{309}{35}a^{9}-\frac{39}{35}a^{8}-\frac{29}{5}a^{7}+\frac{324}{35}a^{6}+\frac{27}{7}a^{5}+\frac{13}{35}a^{4}-\frac{129}{35}a^{3}-\frac{26}{7}a^{2}-\frac{1}{35}a+\frac{96}{35}$, $\frac{2}{35}a^{17}-\frac{2}{35}a^{16}-\frac{4}{7}a^{15}-\frac{8}{35}a^{14}+\frac{26}{35}a^{13}+\frac{13}{7}a^{12}+\frac{38}{35}a^{11}-\frac{61}{35}a^{10}-\frac{128}{35}a^{9}-\frac{43}{35}a^{8}+\frac{7}{5}a^{7}+\frac{158}{35}a^{6}+\frac{19}{7}a^{5}-\frac{79}{35}a^{4}-\frac{83}{35}a^{3}-\frac{10}{7}a^{2}+\frac{33}{35}a+\frac{52}{35}$, $\frac{51}{35}a^{17}-\frac{16}{35}a^{16}-\frac{4}{7}a^{15}-\frac{169}{35}a^{14}-\frac{37}{35}a^{13}+\frac{6}{7}a^{12}+\frac{339}{35}a^{11}+\frac{37}{35}a^{10}+\frac{201}{35}a^{9}-\frac{519}{35}a^{8}-\frac{9}{5}a^{7}-\frac{276}{35}a^{6}+\frac{47}{7}a^{5}+\frac{138}{35}a^{4}+\frac{141}{35}a^{3}-\frac{10}{7}a^{2}-\frac{51}{35}a-\frac{74}{35}$, $\frac{44}{35}a^{17}+\frac{16}{35}a^{16}-\frac{6}{7}a^{15}-\frac{166}{35}a^{14}-\frac{103}{35}a^{13}+\frac{4}{7}a^{12}+\frac{316}{35}a^{11}+\frac{158}{35}a^{10}+\frac{139}{35}a^{9}-\frac{346}{35}a^{8}-\frac{202}{35}a^{7}-\frac{239}{35}a^{6}+\frac{27}{7}a^{5}+\frac{72}{35}a^{4}+\frac{124}{35}a^{3}-\frac{1}{7}a^{2}+\frac{16}{35}a-\frac{3}{5}$ | 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: | \( 107.172918917 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 107.172918917 \cdot 1}{2\cdot\sqrt{10992670662263790889}}\cr\approx \mathstrut & 0.157037145814 \end{aligned}\]
Galois group
$A_4^3.(C_2\times S_4)$ (as 18T776):
A solvable group of order 82944 |
The 65 conjugacy class representatives for $A_4^3.(C_2\times S_4)$ are not computed |
Character table for $A_4^3.(C_2\times S_4)$ is not computed |
Intermediate fields
3.1.23.1, 9.3.3315519667.1 |
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 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 12.0.76256952341.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{7}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | R | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(23\) | 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(379\) | $\Q_{379}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{379}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
\(719\) | $\Q_{719}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{719}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |