Normalized defining polynomial
\( x^{16} + 5x^{14} + 24x^{12} + 115x^{10} + 551x^{8} + 115x^{6} + 24x^{4} + 5x^{2} + 1 \)
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: | \(605165749776000000000000\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\) | 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: | \(30.65\) | 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: | $2\cdot 3^{1/2}5^{3/4}7^{1/2}\approx 30.645530678223075$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(420=2^{2}\cdot 3\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(323,·)$, $\chi_{420}(391,·)$, $\chi_{420}(139,·)$, $\chi_{420}(209,·)$, $\chi_{420}(407,·)$, $\chi_{420}(223,·)$, $\chi_{420}(293,·)$, $\chi_{420}(337,·)$, $\chi_{420}(169,·)$, $\chi_{420}(71,·)$, $\chi_{420}(239,·)$, $\chi_{420}(307,·)$, $\chi_{420}(41,·)$, $\chi_{420}(377,·)$, $\chi_{420}(253,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{551}a^{10}+\frac{115}{551}$, $\frac{1}{551}a^{11}+\frac{115}{551}a$, $\frac{1}{551}a^{12}+\frac{115}{551}a^{2}$, $\frac{1}{551}a^{13}+\frac{115}{551}a^{3}$, $\frac{1}{551}a^{14}+\frac{115}{551}a^{4}$, $\frac{1}{551}a^{15}+\frac{115}{551}a^{5}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1}{551} a^{12} - \frac{2640}{551} a^{2} \) (order $10$) | 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: | $\frac{120}{551}a^{14}+\frac{575}{551}a^{12}+\frac{2760}{551}a^{10}+24a^{8}+115a^{6}+\frac{576}{551}a^{4}+\frac{2760}{551}a^{2}+\frac{575}{551}$, $\frac{5}{551}a^{15}-\frac{12649}{551}a^{5}$, $\frac{436}{551}a^{15}+\frac{115}{551}a^{14}+\frac{2180}{551}a^{13}+\frac{576}{551}a^{12}+\frac{10465}{551}a^{11}+\frac{2760}{551}a^{10}+91a^{9}+24a^{8}+436a^{7}+115a^{6}+\frac{50140}{551}a^{5}+\frac{13225}{551}a^{4}+\frac{10464}{551}a^{3}+\frac{120}{551}a^{2}+\frac{91}{551}a-\frac{527}{551}$, $\frac{1}{29}a^{15}+\frac{235}{551}a^{14}+\frac{1151}{551}a^{12}+\frac{1}{551}a^{11}+\frac{5520}{551}a^{10}+48a^{8}+230a^{6}-\frac{2524}{29}a^{5}+\frac{13801}{551}a^{4}+\frac{2880}{551}a^{2}-\frac{2089}{551}a+\frac{1150}{551}$, $\frac{431}{551}a^{15}-\frac{115}{551}a^{14}+\frac{2180}{551}a^{13}-\frac{576}{551}a^{12}+\frac{10465}{551}a^{11}-\frac{2761}{551}a^{10}+91a^{9}-24a^{8}+436a^{7}-115a^{6}+\frac{62789}{551}a^{5}-\frac{13225}{551}a^{4}+\frac{10464}{551}a^{3}-\frac{120}{551}a^{2}+\frac{91}{551}a+\frac{963}{551}$, $\frac{120}{551}a^{15}-\frac{10}{551}a^{14}+\frac{20}{19}a^{13}-\frac{4}{551}a^{12}+\frac{2761}{551}a^{11}+24a^{9}+115a^{7}+\frac{576}{551}a^{5}+\frac{25298}{551}a^{4}-\frac{341}{19}a^{3}+\frac{10009}{551}a^{2}-\frac{2065}{551}a+2$, $\frac{24}{551}a^{15}+\frac{216}{551}a^{14}+\frac{9}{551}a^{13}+\frac{1147}{551}a^{12}+\frac{1}{551}a^{11}+\frac{5520}{551}a^{10}+48a^{8}+230a^{6}-\frac{60605}{551}a^{5}+\frac{61757}{551}a^{4}-\frac{22658}{551}a^{3}+\frac{12889}{551}a^{2}-\frac{2640}{551}a+\frac{1150}{551}$ (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: | \( 63425.6271299 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 63425.6271299 \cdot 8}{10\cdot\sqrt{605165749776000000000000}}\cr\approx \mathstrut & 0.158436971982 \end{aligned}\] (assuming GRH)
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 | R | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
\(3\) | 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 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(7\) | 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 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |