Normalized defining polynomial
\( x^{18} - 7x^{16} + 3x^{14} + 46x^{12} - 67x^{10} - 41x^{8} + 130x^{6} - 80x^{4} + 17x^{2} - 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[14, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(80421398252924334225227776\) \(\medspace = 2^{18}\cdot 7^{12}\cdot 53^{6}\) | 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: | \(27.49\) | 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: | $2^{3/2}7^{2/3}53^{1/2}\approx 75.34971632639669$ | ||
Ramified primes: | \(2\), \(7\), \(53\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{281}a^{16}-\frac{21}{281}a^{14}+\frac{16}{281}a^{12}+\frac{103}{281}a^{10}-\frac{104}{281}a^{8}+\frac{10}{281}a^{6}-\frac{10}{281}a^{4}+\frac{60}{281}a^{2}+\frac{20}{281}$, $\frac{1}{281}a^{17}-\frac{21}{281}a^{15}+\frac{16}{281}a^{13}+\frac{103}{281}a^{11}-\frac{104}{281}a^{9}+\frac{10}{281}a^{7}-\frac{10}{281}a^{5}+\frac{60}{281}a^{3}+\frac{20}{281}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $15$ | 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: | $\frac{323}{281}a^{17}-\frac{1444}{281}a^{15}-\frac{3543}{281}a^{13}+\frac{10508}{281}a^{11}+\frac{10244}{281}a^{9}-\frac{20093}{281}a^{7}-\frac{6602}{281}a^{5}+\frac{8983}{281}a^{3}-\frac{1408}{281}a$, $\frac{323}{281}a^{16}-\frac{1444}{281}a^{14}-\frac{3543}{281}a^{12}+\frac{10508}{281}a^{10}+\frac{10244}{281}a^{8}-\frac{20093}{281}a^{6}-\frac{6602}{281}a^{4}+\frac{8983}{281}a^{2}-\frac{1408}{281}$, $\frac{323}{281}a^{16}-\frac{1444}{281}a^{14}-\frac{3543}{281}a^{12}+\frac{10508}{281}a^{10}+\frac{10244}{281}a^{8}-\frac{20093}{281}a^{6}-\frac{6602}{281}a^{4}+\frac{8983}{281}a^{2}-\frac{1127}{281}$, $\frac{1901}{281}a^{17}-\frac{11259}{281}a^{15}-\frac{6114}{281}a^{13}+\frac{79188}{281}a^{11}-\frac{43997}{281}a^{9}-\frac{113622}{281}a^{7}+\frac{124300}{281}a^{5}-\frac{36275}{281}a^{3}+\frac{2614}{281}a$, $\frac{1359}{281}a^{16}-\frac{8026}{281}a^{14}-\frac{4389}{281}a^{12}+\frac{55958}{281}a^{10}-\frac{31465}{281}a^{8}-\frac{77735}{281}a^{6}+\frac{88694}{281}a^{4}-\frac{30860}{281}a^{2}+\frac{3295}{281}$, $\frac{133}{281}a^{17}-\frac{1107}{281}a^{15}+\frac{1285}{281}a^{13}+\frac{7517}{281}a^{11}-\frac{15237}{281}a^{9}-\frac{7381}{281}a^{7}+\frac{28175}{281}a^{5}-\frac{12533}{281}a^{3}+\frac{131}{281}a$, $\frac{67}{281}a^{16}-\frac{283}{281}a^{14}-\frac{895}{281}a^{12}+\frac{2405}{281}a^{10}+\frac{3429}{281}a^{8}-\frac{6355}{281}a^{6}-\frac{3480}{281}a^{4}+\frac{5706}{281}a^{2}-\frac{908}{281}$, $\frac{1411}{281}a^{16}-\frac{8275}{281}a^{14}-\frac{4962}{281}a^{12}+\frac{58223}{281}a^{10}-\frac{29848}{281}a^{8}-\frac{83959}{281}a^{6}+\frac{88455}{281}a^{4}-\frac{25773}{281}a^{2}+\frac{1525}{281}$, $a+1$, $\frac{418}{281}a^{17}-\frac{1472}{281}a^{15}-\frac{6800}{281}a^{13}+\frac{11582}{281}a^{11}+\frac{29026}{281}a^{9}-\frac{29821}{281}a^{7}-\frac{33123}{281}a^{5}+\frac{28733}{281}a^{3}-\frac{3723}{281}a+1$, $\frac{884}{281}a^{17}-\frac{442}{281}a^{16}-\frac{5076}{281}a^{15}+\frac{2538}{281}a^{14}-\frac{3559}{281}a^{13}+\frac{1920}{281}a^{12}+\frac{35133}{281}a^{11}-\frac{18269}{281}a^{10}-\frac{15504}{281}a^{9}+\frac{6628}{281}a^{8}-\frac{48203}{281}a^{7}+\frac{29019}{281}a^{6}+\frac{49608}{281}a^{5}-\frac{23399}{281}a^{4}-\frac{19739}{281}a^{3}+\frac{2142}{281}a^{2}+\frac{2787}{281}a+\frac{714}{281}$, $\frac{918}{281}a^{17}+\frac{152}{281}a^{16}-\frac{5228}{281}a^{15}-\frac{1506}{281}a^{14}-\frac{4139}{281}a^{13}+\frac{2713}{281}a^{12}+\frac{37230}{281}a^{11}+\frac{10317}{281}a^{10}-\frac{12577}{281}a^{9}-\frac{26205}{281}a^{8}-\frac{57136}{281}a^{7}-\frac{9439}{281}a^{6}+\frac{45615}{281}a^{5}+\frac{45688}{281}a^{4}-\frac{8145}{281}a^{3}-\frac{20104}{281}a^{2}+\frac{1500}{281}a+\frac{1635}{281}$, $\frac{219}{281}a^{17}-\frac{1198}{281}a^{16}-\frac{1789}{281}a^{15}+\frac{6893}{281}a^{14}+\frac{1818}{281}a^{13}+\frac{4998}{281}a^{12}+\frac{12722}{281}a^{11}-\frac{48929}{281}a^{10}-\frac{22776}{281}a^{9}+\frac{19498}{281}a^{8}-\frac{15794}{281}a^{7}+\frac{73725}{281}a^{6}+\frac{42208}{281}a^{5}-\frac{65295}{281}a^{4}-\frac{14398}{281}a^{3}+\frac{14106}{281}a^{2}+\frac{446}{281}a-\frac{918}{281}$, $\frac{979}{281}a^{17}-\frac{5104}{281}a^{15}-\frac{6816}{281}a^{13}+\frac{36207}{281}a^{11}+\frac{3278}{281}a^{9}-\frac{57931}{281}a^{7}+\frac{23087}{281}a^{5}+\frac{292}{281}a^{3}+\frac{472}{281}a-1$, $\frac{1135}{281}a^{17}-\frac{380}{281}a^{16}-\frac{7537}{281}a^{15}+\frac{2922}{281}a^{14}+\frac{738}{281}a^{13}-\frac{2427}{281}a^{12}+\frac{52275}{281}a^{11}-\frac{19751}{281}a^{10}-\frac{57625}{281}a^{9}+\frac{34743}{281}a^{8}-\frac{65925}{281}a^{7}+\frac{19804}{281}a^{6}+\frac{124373}{281}a^{5}-\frac{65607}{281}a^{4}-\frac{48234}{281}a^{3}+\frac{31152}{281}a^{2}+\frac{2468}{281}a-\frac{2542}{281}$ (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: | \( 4908593.61642 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 4908593.61642 \cdot 1}{2\cdot\sqrt{80421398252924334225227776}}\cr\approx \mathstrut & 0.177019434378 \end{aligned}\] (assuming GRH)
Galois group
$C_2^4:(S_3\times A_4)$ (as 18T268):
A solvable group of order 1152 |
The 24 conjugacy class representatives for $C_2^4:(S_3\times A_4)$ |
Character table for $C_2^4:(S_3\times A_4)$ |
Intermediate fields
\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 12.0.2911171376593984.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.1 | $x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ |
2.12.12.11 | $x^{12} + 28 x^{10} + 40 x^{9} + 356 x^{8} + 896 x^{7} + 2720 x^{6} + 6656 x^{5} + 12464 x^{4} + 19456 x^{3} + 26304 x^{2} + 19840 x + 5824$ | $2$ | $6$ | $12$ | $A_4 \times C_2$ | $[2, 2]^{6}$ | |
\(7\) | 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
\(53\) | 53.3.0.1 | $x^{3} + 3 x + 51$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
53.3.0.1 | $x^{3} + 3 x + 51$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
53.6.3.1 | $x^{6} + 7314 x^{5} + 17831697 x^{4} + 14491896314 x^{3} + 998947242 x^{2} + 44403234204 x + 739971484841$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
53.6.3.1 | $x^{6} + 7314 x^{5} + 17831697 x^{4} + 14491896314 x^{3} + 998947242 x^{2} + 44403234204 x + 739971484841$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |