Normalized defining polynomial
\( x^{16} - 8 x^{14} - 12 x^{13} + 24 x^{12} + 80 x^{11} + 38 x^{10} - 152 x^{9} - 287 x^{8} - 120 x^{7} + \cdots + 1 \)
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: | \(103670069459943424\) \(\medspace = 2^{32}\cdot 17^{6}\) | 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: | \(11.57\) | 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: | $2^{2}17^{1/2}\approx 16.492422502470642$ | ||
Ramified primes: | \(2\), \(17\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{13}+\frac{1}{8}a^{11}+\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}-\frac{3}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}+\frac{1}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{3}{8}a-\frac{1}{8}$, $\frac{1}{1784}a^{15}+\frac{81}{1784}a^{14}+\frac{43}{892}a^{13}+\frac{41}{1784}a^{12}+\frac{1}{8}a^{11}+\frac{10}{223}a^{10}-\frac{395}{1784}a^{9}+\frac{411}{1784}a^{8}+\frac{3}{8}a^{7}-\frac{283}{892}a^{6}-\frac{327}{1784}a^{5}-\frac{147}{1784}a^{4}-\frac{65}{892}a^{3}+\frac{605}{1784}a^{2}+\frac{457}{1784}a+\frac{117}{892}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{17183}{1784} a^{15} - \frac{1987}{446} a^{14} - \frac{133437}{1784} a^{13} - \frac{144681}{1784} a^{12} + 267 a^{11} + \frac{1152309}{1784} a^{10} + \frac{131513}{1784} a^{9} - \frac{1325499}{892} a^{8} - \frac{8327}{4} a^{7} - \frac{403063}{1784} a^{6} + \frac{4426863}{1784} a^{5} + \frac{761213}{223} a^{4} + \frac{4220499}{1784} a^{3} + \frac{1706743}{1784} a^{2} + \frac{191623}{892} a + \frac{38265}{1784} \) (order $8$) | 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: | $\frac{381}{223}a^{15}-\frac{941}{446}a^{14}-\frac{10541}{892}a^{13}-\frac{1104}{223}a^{12}+\frac{209}{4}a^{11}+\frac{33085}{446}a^{10}-\frac{22017}{446}a^{9}-\frac{102713}{446}a^{8}-\frac{739}{4}a^{7}+\frac{53733}{446}a^{6}+\frac{79681}{223}a^{5}+\frac{141983}{446}a^{4}+\frac{134373}{892}a^{3}+\frac{6836}{223}a^{2}-\frac{2191}{892}a-\frac{1207}{446}$, $\frac{12245}{1784}a^{15}-\frac{1743}{446}a^{14}-\frac{93819}{1784}a^{13}-\frac{93365}{1784}a^{12}+\frac{387}{2}a^{11}+\frac{780461}{1784}a^{10}+\frac{25963}{1784}a^{9}-\frac{928891}{892}a^{8}-\frac{5495}{4}a^{7}-\frac{110215}{1784}a^{6}+\frac{3064093}{1784}a^{5}+\frac{502898}{223}a^{4}+\frac{2722085}{1784}a^{3}+\frac{1089743}{1784}a^{2}+\frac{124547}{892}a+\frac{26317}{1784}$, $\frac{8223}{1784}a^{15}-\frac{9181}{1784}a^{14}-\frac{29079}{892}a^{13}-\frac{30807}{1784}a^{12}+\frac{1123}{8}a^{11}+\frac{194005}{892}a^{10}-\frac{201905}{1784}a^{9}-\frac{1149031}{1784}a^{8}-\frac{4519}{8}a^{7}+\frac{64198}{223}a^{6}+\frac{1798731}{1784}a^{5}+\frac{1684867}{1784}a^{4}+\frac{424851}{892}a^{3}+\frac{239733}{1784}a^{2}+\frac{39163}{1784}a+\frac{815}{446}$, $\frac{9037}{1784}a^{15}-\frac{6133}{1784}a^{14}-\frac{16997}{446}a^{13}-\frac{62103}{1784}a^{12}+\frac{1155}{8}a^{11}+\frac{272503}{892}a^{10}-\frac{23023}{1784}a^{9}-\frac{1339419}{1784}a^{8}-\frac{7499}{8}a^{7}+\frac{4295}{446}a^{6}+\frac{2178361}{1784}a^{5}+\frac{2760935}{1784}a^{4}+\frac{460595}{446}a^{3}+\frac{736217}{1784}a^{2}+\frac{171659}{1784}a+\frac{2586}{223}$, $\frac{4981}{1784}a^{15}-\frac{3291}{1784}a^{14}-\frac{18629}{892}a^{13}-\frac{35281}{1784}a^{12}+\frac{625}{8}a^{11}+\frac{151295}{892}a^{10}+\frac{2041}{1784}a^{9}-\frac{727821}{1784}a^{8}-\frac{4253}{8}a^{7}-\frac{5808}{223}a^{6}+\frac{1182797}{1784}a^{5}+\frac{1581641}{1784}a^{4}+\frac{550841}{892}a^{3}+\frac{451235}{1784}a^{2}+\frac{98053}{1784}a+\frac{1023}{223}$, $\frac{9935}{1784}a^{15}-\frac{4273}{892}a^{14}-\frac{73047}{1784}a^{13}-\frac{55167}{1784}a^{12}+\frac{655}{4}a^{11}+\frac{545709}{1784}a^{10}-\frac{120391}{1784}a^{9}-\frac{180527}{223}a^{8}-885a^{7}+\frac{289189}{1784}a^{6}+\frac{2323579}{1784}a^{5}+\frac{1307217}{892}a^{4}+\frac{1575561}{1784}a^{3}+\frac{559217}{1784}a^{2}+\frac{28269}{446}a+\frac{11165}{1784}$, $\frac{711}{446}a^{15}-\frac{2225}{1784}a^{14}-\frac{21901}{1784}a^{13}-\frac{8375}{892}a^{12}+\frac{399}{8}a^{11}+\frac{165749}{1784}a^{10}-\frac{24483}{892}a^{9}-\frac{458793}{1784}a^{8}-\frac{2045}{8}a^{7}+\frac{172735}{1784}a^{6}+\frac{96159}{223}a^{5}+\frac{722131}{1784}a^{4}+\frac{314667}{1784}a^{3}+\frac{21607}{892}a^{2}-\frac{13539}{1784}a-\frac{4173}{1784}$ | 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: | \( 312.978531942 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 312.978531942 \cdot 1}{8\cdot\sqrt{103670069459943424}}\cr\approx \mathstrut & 0.295145940507 \end{aligned}\]
Galois group
$C_2^2\wr C_2$ (as 16T46):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_2^2\wr C_2$ |
Character table for $C_2^2\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), 4.0.4352.1, 4.0.1088.1, 4.0.4352.2, 4.0.272.1, 4.0.1088.2, \(\Q(\zeta_{8})\), 4.4.4352.1, 8.0.18939904.2, 8.0.18939904.3, 8.0.18939904.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
\(17\) | 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |