Normalized defining polynomial
\( x^{32} + 449x^{16} + 65536 \)
Invariants
Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(11308580273943039971119355619759072515285051093549056\) \(\medspace = 2^{128}\cdot 7^{16}\) | 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: | \(42.33\) | 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: | $2^{4}7^{1/2}\approx 42.33202097703345$ | ||
Ramified primes: | \(2\), \(7\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(224=2^{5}\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{224}(1,·)$, $\chi_{224}(139,·)$, $\chi_{224}(13,·)$, $\chi_{224}(15,·)$, $\chi_{224}(153,·)$, $\chi_{224}(155,·)$, $\chi_{224}(29,·)$, $\chi_{224}(69,·)$, $\chi_{224}(27,·)$, $\chi_{224}(167,·)$, $\chi_{224}(41,·)$, $\chi_{224}(43,·)$, $\chi_{224}(181,·)$, $\chi_{224}(183,·)$, $\chi_{224}(57,·)$, $\chi_{224}(195,·)$, $\chi_{224}(197,·)$, $\chi_{224}(71,·)$, $\chi_{224}(55,·)$, $\chi_{224}(141,·)$, $\chi_{224}(209,·)$, $\chi_{224}(83,·)$, $\chi_{224}(85,·)$, $\chi_{224}(223,·)$, $\chi_{224}(97,·)$, $\chi_{224}(99,·)$, $\chi_{224}(111,·)$, $\chi_{224}(113,·)$, $\chi_{224}(211,·)$, $\chi_{224}(169,·)$, $\chi_{224}(125,·)$, $\chi_{224}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{93}a^{16}-\frac{8}{93}$, $\frac{1}{186}a^{17}+\frac{85}{186}a$, $\frac{1}{372}a^{18}+\frac{85}{372}a^{2}$, $\frac{1}{744}a^{19}-\frac{287}{744}a^{3}$, $\frac{1}{1488}a^{20}-\frac{287}{1488}a^{4}$, $\frac{1}{2976}a^{21}-\frac{287}{2976}a^{5}$, $\frac{1}{5952}a^{22}+\frac{2689}{5952}a^{6}$, $\frac{1}{11904}a^{23}-\frac{3263}{11904}a^{7}$, $\frac{1}{23808}a^{24}+\frac{8641}{23808}a^{8}$, $\frac{1}{47616}a^{25}+\frac{8641}{47616}a^{9}$, $\frac{1}{95232}a^{26}+\frac{8641}{95232}a^{10}$, $\frac{1}{190464}a^{27}+\frac{8641}{190464}a^{11}$, $\frac{1}{380928}a^{28}+\frac{8641}{380928}a^{12}$, $\frac{1}{761856}a^{29}+\frac{8641}{761856}a^{13}$, $\frac{1}{1523712}a^{30}-\frac{753215}{1523712}a^{14}$, $\frac{1}{3047424}a^{31}-\frac{753215}{3047424}a^{15}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{136}$, which has order $136$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{7}{11904} a^{23} + \frac{967}{11904} a^{7} \) (order $32$) | 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: | $\frac{17}{47616}a^{25}+\frac{1}{2976}a^{21}-\frac{1}{186}a^{17}+\frac{4049}{47616}a^{9}-\frac{287}{2976}a^{5}-\frac{271}{186}a$, $\frac{91}{1523712}a^{30}-\frac{1}{7936}a^{24}+\frac{1}{496}a^{20}+\frac{24475}{1523712}a^{14}-\frac{705}{7936}a^{8}+\frac{209}{496}a^{4}$, $\frac{91}{1523712}a^{30}-\frac{1}{7936}a^{24}+\frac{1}{744}a^{19}+\frac{24475}{1523712}a^{14}-\frac{705}{7936}a^{8}+\frac{457}{744}a^{3}$, $\frac{1}{761856}a^{29}-\frac{1}{744}a^{19}+\frac{8641}{761856}a^{13}-\frac{457}{744}a^{3}-1$, $\frac{1}{372}a^{18}-\frac{1}{186}a^{17}+\frac{85}{372}a^{2}-\frac{271}{186}a+1$, $\frac{1}{372}a^{18}+\frac{1}{186}a^{17}+\frac{85}{372}a^{2}+\frac{271}{186}a+1$, $\frac{11}{95232}a^{26}-\frac{1}{7936}a^{24}+\frac{1}{186}a^{17}-\frac{181}{95232}a^{10}-\frac{705}{7936}a^{8}+\frac{271}{186}a$, $\frac{11}{95232}a^{26}+\frac{1}{7936}a^{24}+\frac{1}{372}a^{18}-\frac{1}{93}a^{16}-\frac{181}{95232}a^{10}+\frac{705}{7936}a^{8}+\frac{457}{372}a^{2}-\frac{178}{93}$, $\frac{11}{95232}a^{26}-\frac{17}{23808}a^{24}-\frac{1}{186}a^{18}-\frac{181}{95232}a^{10}-\frac{4049}{23808}a^{8}-\frac{271}{186}a^{2}+1$, $\frac{1}{761856}a^{30}+\frac{11}{95232}a^{26}+\frac{1}{7936}a^{24}+\frac{1}{1488}a^{20}-\frac{1}{372}a^{18}+\frac{8641}{761856}a^{14}-\frac{181}{95232}a^{10}+\frac{705}{7936}a^{8}-\frac{287}{1488}a^{4}-\frac{85}{372}a^{2}+1$, $\frac{23}{190464}a^{27}-\frac{17}{47616}a^{25}+\frac{5}{5952}a^{23}-\frac{5}{2976}a^{21}+\frac{1}{744}a^{19}-\frac{1}{186}a^{17}+\frac{8279}{190464}a^{11}-\frac{4049}{47616}a^{9}+\frac{1541}{5952}a^{7}-\frac{1541}{2976}a^{5}+\frac{457}{744}a^{3}-\frac{271}{186}a$, $\frac{1}{32768}a^{31}-\frac{11}{95232}a^{26}-\frac{17}{47616}a^{25}-\frac{1}{744}a^{20}+\frac{449}{32768}a^{15}+\frac{181}{95232}a^{10}-\frac{4049}{47616}a^{9}-\frac{457}{744}a^{4}$, $\frac{91}{1523712}a^{30}+\frac{1}{380928}a^{28}+\frac{1}{496}a^{21}+\frac{1}{744}a^{19}+\frac{24475}{1523712}a^{14}+\frac{8641}{380928}a^{12}+\frac{209}{496}a^{5}+\frac{457}{744}a^{3}$, $\frac{23}{190464}a^{27}+\frac{17}{47616}a^{26}+\frac{11}{47616}a^{25}-\frac{1}{7936}a^{24}+\frac{8279}{190464}a^{11}+\frac{4049}{47616}a^{10}-\frac{181}{47616}a^{9}-\frac{705}{7936}a^{8}$, $\frac{91}{1523712}a^{30}+\frac{7}{11904}a^{23}+\frac{1}{744}a^{20}-\frac{1}{186}a^{17}+\frac{24475}{1523712}a^{14}+\frac{967}{11904}a^{7}+\frac{457}{744}a^{4}-\frac{85}{186}a$ (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: | \( 2002471224006.0598 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 2002471224006.0598 \cdot 136}{32\cdot\sqrt{11308580273943039971119355619759072515285051093549056}}\cr\approx \mathstrut & 0.472203351351673 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times C_8$ (as 32T37):
An abelian group of order 32 |
The 32 conjugacy class representatives for $C_2^2\times C_8$ |
Character table for $C_2^2\times C_8$ |
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 | ${\href{/padicField/3.8.0.1}{8} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{4}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{16}$ | ${\href{/padicField/19.8.0.1}{8} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{16}$ | ${\href{/padicField/53.8.0.1}{8} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.64.2 | $x^{16} + 16 x^{13} + 20 x^{12} + 22 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 2$ | $16$ | $1$ | $64$ | $C_8\times C_2$ | $[2, 3, 4, 5]$ |
2.16.64.2 | $x^{16} + 16 x^{13} + 20 x^{12} + 22 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 2$ | $16$ | $1$ | $64$ | $C_8\times C_2$ | $[2, 3, 4, 5]$ | |
\(7\) | 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |