Normalized defining polynomial
\( x^{18} - x^{16} - 65x^{14} - 297x^{12} - 549x^{10} - 433x^{8} - 77x^{6} + 48x^{4} + 5x^{2} - 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(75613185918270483380568064\) \(\medspace = 2^{18}\cdot 19^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(27.40\) | 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))
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Galois root discriminant: | $2^{3/2}19^{8/9}\approx 38.74492825271069$ | ||
Ramified primes: | \(2\), \(19\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7067}a^{16}-\frac{3323}{7067}a^{14}+\frac{287}{7067}a^{12}+\frac{334}{7067}a^{10}-\frac{578}{7067}a^{8}-\frac{2541}{7067}a^{6}+\frac{3127}{7067}a^{4}+\frac{644}{7067}a^{2}+\frac{1938}{7067}$, $\frac{1}{7067}a^{17}-\frac{3323}{7067}a^{15}+\frac{287}{7067}a^{13}+\frac{334}{7067}a^{11}-\frac{578}{7067}a^{9}-\frac{2541}{7067}a^{7}+\frac{3127}{7067}a^{5}+\frac{644}{7067}a^{3}+\frac{1938}{7067}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $11$ | 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)
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Fundamental units: | $\frac{16137}{7067}a^{17}-\frac{34190}{7067}a^{15}-\frac{1008147}{7067}a^{13}-\frac{3670136}{7067}a^{11}-\frac{4910311}{7067}a^{9}-\frac{2064947}{7067}a^{7}+\frac{312967}{7067}a^{5}+\frac{138011}{7067}a^{3}+\frac{23232}{7067}a$, $\frac{17166}{7067}a^{16}-\frac{40196}{7067}a^{14}-\frac{1066174}{7067}a^{12}-\frac{3658599}{7067}a^{10}-\frac{4247147}{7067}a^{8}-\frac{764518}{7067}a^{6}+\frac{993597}{7067}a^{4}+\frac{37451}{7067}a^{2}-\frac{24929}{7067}$, $\frac{6073}{7067}a^{16}-\frac{11361}{7067}a^{14}-\frac{384216}{7067}a^{12}-\frac{1469783}{7067}a^{10}-\frac{2096794}{7067}a^{8}-\frac{986545}{7067}a^{6}+\frac{107247}{7067}a^{4}+\frac{52430}{7067}a^{2}-\frac{11215}{7067}$, $\frac{12121}{7067}a^{16}-\frac{24451}{7067}a^{14}-\frac{761473}{7067}a^{12}-\frac{2827777}{7067}a^{10}-\frac{3875257}{7067}a^{8}-\frac{1697555}{7067}a^{6}+\frac{199922}{7067}a^{4}+\frac{67559}{7067}a^{2}+\frac{6857}{7067}$, $\frac{4922}{7067}a^{17}-\frac{16902}{7067}a^{15}-\frac{290533}{7067}a^{13}-\frac{723497}{7067}a^{11}-\frac{223059}{7067}a^{9}+\frac{694354}{7067}a^{7}+\frac{189977}{7067}a^{5}-\frac{293062}{7067}a^{3}+\frac{19587}{7067}a$, $\frac{4977}{7067}a^{16}-\frac{8858}{7067}a^{14}-\frac{317150}{7067}a^{12}-\frac{1228085}{7067}a^{10}-\frac{1745986}{7067}a^{8}-\frac{759863}{7067}a^{6}+\frac{149952}{7067}a^{4}+\frac{46239}{7067}a^{2}-\frac{8096}{7067}$, $\frac{10942}{7067}a^{16}-\frac{28819}{7067}a^{14}-\frac{668759}{7067}a^{12}-\frac{2147379}{7067}a^{10}-\frac{2211482}{7067}a^{8}-\frac{72714}{7067}a^{6}+\frac{732188}{7067}a^{4}+\frac{57385}{7067}a^{2}-\frac{23672}{7067}$, $\frac{10089}{7067}a^{16}-\frac{21100}{7067}a^{14}-\frac{630890}{7067}a^{12}-\frac{2312142}{7067}a^{10}-\frac{3131848}{7067}a^{8}-\frac{1353937}{7067}a^{6}+\frac{220292}{7067}a^{4}+\frac{122882}{7067}a^{2}-\frac{1907}{7067}$, $\frac{24}{191}a^{17}-\frac{16956}{7067}a^{16}-\frac{487}{191}a^{15}+\frac{27865}{7067}a^{14}-\frac{561}{191}a^{13}+\frac{1076975}{7067}a^{12}+\frac{21768}{191}a^{11}+\frac{4357702}{7067}a^{10}+\frac{90987}{191}a^{9}+\frac{6959634}{7067}a^{8}+\frac{125814}{191}a^{7}+\frac{4548845}{7067}a^{6}+\frac{49836}{191}a^{5}+\frac{687788}{7067}a^{4}-\frac{12239}{191}a^{3}-\frac{227293}{7067}a^{2}-\frac{3530}{191}a-\frac{41580}{7067}$, $\frac{2744}{7067}a^{17}-\frac{2919}{7067}a^{16}+\frac{5185}{7067}a^{15}+\frac{3913}{7067}a^{14}-\frac{194785}{7067}a^{13}+\frac{186962}{7067}a^{12}-\frac{1309609}{7067}a^{11}+\frac{805938}{7067}a^{10}-\frac{3338648}{7067}a^{9}+\frac{1418636}{7067}a^{8}-\frac{3771153}{7067}a^{7}+\frac{1134616}{7067}a^{6}-\frac{1560657}{7067}a^{5}+\frac{377402}{7067}a^{4}+\frac{49855}{7067}a^{3}+\frac{42388}{7067}a^{2}+\frac{38823}{7067}a-\frac{3422}{7067}$, $\frac{12752}{7067}a^{17}-\frac{2232}{7067}a^{16}+\frac{5903}{7067}a^{15}-\frac{3414}{7067}a^{14}-\frac{870123}{7067}a^{13}+\frac{157987}{7067}a^{12}-\frac{4949133}{7067}a^{11}+\frac{1014195}{7067}a^{10}-\frac{11137367}{7067}a^{9}+\frac{2449084}{7067}a^{8}-\frac{10806080}{7067}a^{7}+\frac{2434826}{7067}a^{6}-\frac{2795042}{7067}a^{5}+\frac{490355}{7067}a^{4}+\frac{1343164}{7067}a^{3}-\frac{476296}{7067}a^{2}+\frac{303958}{7067}a-\frac{92483}{7067}$ | 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: | \( 662381.694109 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 662381.694109 \cdot 1}{2\cdot\sqrt{75613185918270483380568064}}\cr\approx \mathstrut & 0.149981862157 \end{aligned}\]
Galois group
$C_2^2:C_9$ (as 18T7):
A solvable group of order 36 |
The 12 conjugacy class representatives for $C_2^2 : C_9$ |
Character table for $C_2^2 : C_9$ |
Intermediate fields
3.3.361.1, 6.2.8340544.2, \(\Q(\zeta_{19})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 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 | R | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.18.18.89 | $x^{18} + 18 x^{17} + 66 x^{16} - 272 x^{15} - 1608 x^{14} + 7008 x^{13} + 83536 x^{12} + 346688 x^{11} + 922880 x^{10} + 2307136 x^{9} + 7066496 x^{8} + 20902656 x^{7} + 47520384 x^{6} + 81117696 x^{5} + 108969728 x^{4} + 117408768 x^{3} + 95319808 x^{2} + 50121216 x + 12416512$ | $2$ | $9$ | $18$ | $C_2^2 : C_9$ | $[2, 2]^{9}$ |
\(19\) | 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |