Normalized defining polynomial
\( x^{9} - x^{8} - 765 x^{7} - 17688 x^{6} - 209034 x^{5} - 1507947 x^{4} - 6871473 x^{3} - 19300827 x^{2} + \cdots - 20768489 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-287752047109343823899619790336\) \(\medspace = -\,2^{9}\cdot 7^{7}\cdot 13^{7}\cdot 19^{7}\cdot 23^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(1875.96\) | 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: | $2^{3/2}7^{5/6}13^{5/6}19^{5/6}23^{1/2}\approx 6769.7309043987025$ | ||
Ramified primes: | \(2\), \(7\), \(13\), \(19\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-79534}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{46}a^{6}-\frac{8}{23}a^{5}+\frac{1}{46}a^{4}-\frac{5}{23}a^{3}+\frac{2}{23}a^{2}+\frac{11}{23}a-\frac{7}{46}$, $\frac{1}{46}a^{7}+\frac{21}{46}a^{5}+\frac{3}{23}a^{4}-\frac{9}{23}a^{3}-\frac{3}{23}a^{2}-\frac{1}{2}a-\frac{10}{23}$, $\frac{1}{46}a^{8}+\frac{10}{23}a^{5}+\frac{7}{46}a^{4}+\frac{10}{23}a^{3}-\frac{15}{46}a^{2}-\frac{11}{23}a+\frac{9}{46}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}\times C_{3}\times C_{6789}$, which has order $183303$ (assuming GRH)
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: | $a^{8}-6a^{7}-735a^{6}-14013a^{5}-138969a^{4}-813102a^{3}-2805963a^{2}-5271012a-4153698$, $a+4$, $\frac{25}{46}a^{8}-\frac{157}{46}a^{7}-\frac{9150}{23}a^{6}-\frac{345533}{46}a^{5}-\frac{147771}{2}a^{4}-\frac{9855510}{23}a^{3}-\frac{67368747}{46}a^{2}-\frac{125273991}{46}a-\frac{97705337}{46}$, $\frac{53}{46}a^{8}-\frac{135}{23}a^{7}-\frac{39437}{46}a^{6}-16870a^{5}-\frac{3951641}{23}a^{4}-\frac{23795686}{23}a^{3}-\frac{169577131}{46}a^{2}-\frac{164892463}{23}a-\frac{134761461}{23}$, $\frac{67\!\cdots\!39}{23}a^{8}-\frac{25\!\cdots\!23}{23}a^{7}-\frac{51\!\cdots\!29}{23}a^{6}-45\!\cdots\!53a^{5}-\frac{11\!\cdots\!11}{23}a^{4}-\frac{69\!\cdots\!15}{23}a^{3}-\frac{25\!\cdots\!51}{23}a^{2}-\frac{51\!\cdots\!60}{23}a-\frac{43\!\cdots\!03}{23}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 960727.9900671571 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 960727.9900671571 \cdot 183303}{2\cdot\sqrt{287752047109343823899619790336}}\cr\approx \mathstrut & 0.325731982085022 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times S_3$ (as 9T4):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3\times C_3$ |
Character table for $S_3\times C_3$ |
Intermediate fields
3.3.2989441.1, 3.1.318136.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 6 sibling: | data not computed |
Minimal sibling: | 6.0.96256138558795381444096.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | R | R | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.9.1 | $x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(7\) | 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.6.5.4 | $x^{6} + 14$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(13\) | 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
13.6.5.4 | $x^{6} + 26$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(19\) | 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
19.6.5.4 | $x^{6} + 76$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(23\) | 23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |