Normalized defining polynomial
\( x^{16} + 8x^{14} + 45x^{12} + 128x^{10} + 264x^{8} + 212x^{6} + 125x^{4} + 12x^{2} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(6879707136000000000000\) \(\medspace = 2^{32}\cdot 3^{8}\cdot 5^{12}\) | 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: | \(23.17\) | 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: | $2^{2}3^{1/2}5^{3/4}\approx 23.165843705765383$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(120=2^{3}\cdot 3\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{120}(1,·)$, $\chi_{120}(67,·)$, $\chi_{120}(7,·)$, $\chi_{120}(43,·)$, $\chi_{120}(83,·)$, $\chi_{120}(23,·)$, $\chi_{120}(89,·)$, $\chi_{120}(29,·)$, $\chi_{120}(101,·)$, $\chi_{120}(103,·)$, $\chi_{120}(41,·)$, $\chi_{120}(107,·)$, $\chi_{120}(109,·)$, $\chi_{120}(47,·)$, $\chi_{120}(49,·)$, $\chi_{120}(61,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19}a^{10}+\frac{8}{19}a^{8}+\frac{7}{19}a^{6}+\frac{7}{19}a^{4}-\frac{1}{19}a^{2}-\frac{8}{19}$, $\frac{1}{19}a^{11}+\frac{8}{19}a^{9}+\frac{7}{19}a^{7}+\frac{7}{19}a^{5}-\frac{1}{19}a^{3}-\frac{8}{19}a$, $\frac{1}{76}a^{12}+\frac{2}{19}a^{6}-\frac{31}{76}$, $\frac{1}{76}a^{13}+\frac{2}{19}a^{7}-\frac{31}{76}a$, $\frac{1}{836}a^{14}+\frac{1}{209}a^{12}-\frac{1}{209}a^{10}-\frac{63}{209}a^{8}-\frac{56}{209}a^{6}-\frac{64}{209}a^{4}+\frac{49}{836}a^{2}+\frac{53}{209}$, $\frac{1}{836}a^{15}+\frac{1}{209}a^{13}-\frac{1}{209}a^{11}-\frac{63}{209}a^{9}-\frac{56}{209}a^{7}-\frac{64}{209}a^{5}+\frac{49}{836}a^{3}+\frac{53}{209}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{4}$, which has order $8$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{85}{836} a^{14} - \frac{335}{418} a^{12} - \frac{938}{209} a^{10} - \frac{2620}{209} a^{8} - \frac{5360}{209} a^{6} - \frac{4020}{209} a^{4} - \frac{10105}{836} a^{2} - \frac{67}{418} \) (order $6$) | 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: | $\frac{45}{836}a^{14}+\frac{189}{418}a^{12}+\frac{538}{209}a^{10}+\frac{1620}{209}a^{8}+\frac{180}{11}a^{6}+\frac{3291}{209}a^{4}+\frac{6561}{836}a^{2}+\frac{315}{418}$, $\frac{43}{836}a^{15}+\frac{315}{836}a^{13}+\frac{430}{209}a^{11}+\frac{1075}{209}a^{9}+\frac{2025}{209}a^{7}+\frac{559}{209}a^{5}+\frac{215}{836}a^{3}-\frac{2929}{836}a$, $\frac{31}{209}a^{15}+\frac{457}{418}a^{13}+\frac{1240}{209}a^{11}+\frac{3100}{209}a^{9}+\frac{5739}{209}a^{7}+\frac{1612}{209}a^{5}+\frac{155}{209}a^{3}-\frac{2619}{418}a$, $\frac{43}{836}a^{15}+\frac{85}{836}a^{14}+\frac{315}{836}a^{13}+\frac{335}{418}a^{12}+\frac{430}{209}a^{11}+\frac{938}{209}a^{10}+\frac{1075}{209}a^{9}+\frac{2620}{209}a^{8}+\frac{2025}{209}a^{7}+\frac{5360}{209}a^{6}+\frac{559}{209}a^{5}+\frac{4020}{209}a^{4}+\frac{215}{836}a^{3}+\frac{10105}{836}a^{2}-\frac{3765}{836}a+\frac{485}{418}$, $\frac{45}{836}a^{15}+\frac{39}{418}a^{14}+\frac{189}{418}a^{13}+\frac{155}{209}a^{12}+\frac{538}{209}a^{11}+\frac{868}{209}a^{10}+\frac{1620}{209}a^{9}+\frac{2445}{209}a^{8}+\frac{180}{11}a^{7}+\frac{4960}{209}a^{6}+\frac{3291}{209}a^{5}+\frac{3720}{209}a^{4}+\frac{6561}{836}a^{3}+\frac{177}{22}a^{2}+\frac{315}{418}a+\frac{31}{209}$, $\frac{163}{836}a^{15}-\frac{5}{418}a^{14}+\frac{645}{418}a^{13}-\frac{73}{836}a^{12}+\frac{1806}{209}a^{11}-\frac{100}{209}a^{10}+\frac{5065}{209}a^{9}-\frac{250}{209}a^{8}+\frac{10320}{209}a^{7}-\frac{485}{209}a^{6}+\frac{7740}{209}a^{5}-\frac{130}{209}a^{4}+\frac{16831}{836}a^{3}-\frac{25}{418}a^{2}+\frac{129}{418}a+\frac{751}{836}$, $\frac{31}{209}a^{15}-\frac{215}{836}a^{14}+\frac{457}{418}a^{13}-\frac{859}{418}a^{12}+\frac{1240}{209}a^{11}-\frac{2414}{209}a^{10}+\frac{3100}{209}a^{9}-\frac{6860}{209}a^{8}+\frac{5739}{209}a^{7}-\frac{14140}{209}a^{6}+\frac{1612}{209}a^{5}-\frac{11331}{209}a^{4}+\frac{155}{209}a^{3}-\frac{1409}{44}a^{2}-\frac{3037}{418}a-\frac{1285}{418}$ | 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: | \( 7114.13535725 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7114.13535725 \cdot 8}{6\cdot\sqrt{6879707136000000000000}}\cr\approx \mathstrut & 0.277788864156 \end{aligned}\]
Galois group
$C_2^2\times C_4$ (as 16T2):
An abelian group of order 16 |
The 16 conjugacy class representatives for $C_4\times C_2^2$ |
Character table for $C_4\times C_2^2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.3 | $x^{8} + 8 x^{7} + 16 x^{6} + 8 x^{5} + 36 x^{4} - 32 x^{3} + 88 x^{2} - 32 x + 124$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ |
2.8.16.3 | $x^{8} + 8 x^{7} + 16 x^{6} + 8 x^{5} + 36 x^{4} - 32 x^{3} + 88 x^{2} - 32 x + 124$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ | |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |