Normalized defining polynomial
\( x^{16} - 23x^{14} + 190x^{12} - 744x^{10} + 1505x^{8} - 1600x^{6} + 872x^{4} - 216x^{2} + 16 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(176556947862350388527104\) \(\medspace = 2^{20}\cdot 17^{14}\) | 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: | \(28.37\) | 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: | $2^{2}17^{7/8}\approx 47.72025959461256$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{26}a^{10}-\frac{1}{2}a^{8}-\frac{6}{13}a^{4}+\frac{7}{26}a^{2}-\frac{2}{13}$, $\frac{1}{26}a^{11}-\frac{1}{2}a^{9}-\frac{6}{13}a^{5}+\frac{7}{26}a^{3}-\frac{2}{13}a$, $\frac{1}{52}a^{12}-\frac{1}{52}a^{10}-\frac{3}{13}a^{6}-\frac{7}{52}a^{4}+\frac{1}{26}a^{2}+\frac{1}{13}$, $\frac{1}{52}a^{13}-\frac{1}{52}a^{11}-\frac{3}{13}a^{7}-\frac{7}{52}a^{5}+\frac{1}{26}a^{3}+\frac{1}{13}a$, $\frac{1}{104}a^{14}-\frac{1}{104}a^{12}+\frac{5}{13}a^{8}-\frac{7}{104}a^{6}-\frac{25}{52}a^{4}+\frac{1}{26}a^{2}$, $\frac{1}{104}a^{15}-\frac{1}{104}a^{13}+\frac{5}{13}a^{9}-\frac{7}{104}a^{7}-\frac{25}{52}a^{5}+\frac{1}{26}a^{3}$
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: | $\frac{5}{8}a^{14}-\frac{1435}{104}a^{12}+\frac{5511}{52}a^{10}-\frac{733}{2}a^{8}+\frac{62161}{104}a^{6}-\frac{11303}{26}a^{4}+\frac{3297}{26}a^{2}-\frac{142}{13}$, $\frac{75}{104}a^{14}-\frac{1665}{104}a^{12}+\frac{6459}{52}a^{10}-\frac{5683}{13}a^{8}+\frac{76483}{104}a^{6}-\frac{14693}{26}a^{4}+\frac{2298}{13}a^{2}-\frac{180}{13}$, $\frac{47}{104}a^{14}-\frac{1049}{104}a^{12}+\frac{4109}{52}a^{10}-\frac{3678}{13}a^{8}+\frac{51111}{104}a^{6}-\frac{5223}{13}a^{4}+\frac{1810}{13}a^{2}-\frac{170}{13}$, $a^{14}-\frac{1151}{52}a^{12}+\frac{8885}{52}a^{10}-\frac{1193}{2}a^{8}+\frac{12878}{13}a^{6}-\frac{38731}{52}a^{4}+\frac{2910}{13}a^{2}-\frac{201}{13}$, $\frac{47}{104}a^{14}-\frac{1049}{104}a^{12}+\frac{4109}{52}a^{10}-\frac{3678}{13}a^{8}+\frac{51111}{104}a^{6}-\frac{5223}{13}a^{4}+\frac{1810}{13}a^{2}-\frac{157}{13}$, $\frac{5}{13}a^{14}-\frac{225}{26}a^{12}+\frac{894}{13}a^{10}-\frac{6555}{26}a^{8}+\frac{5909}{13}a^{6}-\frac{10149}{26}a^{4}+\frac{277}{2}a^{2}-\frac{170}{13}$, $\frac{105}{104}a^{14}-\frac{2335}{104}a^{12}+\frac{699}{4}a^{10}-\frac{8042}{13}a^{8}+\frac{109433}{104}a^{6}-\frac{10726}{13}a^{4}+\frac{3436}{13}a^{2}-21$, $\frac{9}{8}a^{15}+\frac{41}{104}a^{14}-\frac{2631}{104}a^{13}-\frac{933}{104}a^{12}+\frac{803}{4}a^{11}+\frac{1889}{26}a^{10}-\frac{1467}{2}a^{9}-\frac{3565}{13}a^{8}+\frac{136573}{104}a^{7}+\frac{53681}{104}a^{6}-\frac{29015}{26}a^{5}-\frac{24471}{52}a^{4}+\frac{10327}{26}a^{3}+\frac{4653}{26}a^{2}-36a-\frac{203}{13}$, $\frac{53}{104}a^{15}+\frac{23}{104}a^{14}-\frac{1189}{104}a^{13}-\frac{509}{104}a^{12}+\frac{1175}{13}a^{11}+\frac{1961}{52}a^{10}-\frac{4272}{13}a^{9}-\frac{3397}{26}a^{8}+\frac{60893}{104}a^{7}+\frac{21999}{104}a^{6}-\frac{25925}{52}a^{5}-\frac{3817}{26}a^{4}+\frac{4795}{26}a^{3}+\frac{879}{26}a^{2}-\frac{280}{13}a+\frac{2}{13}$, $\frac{233}{104}a^{15}-\frac{5}{8}a^{14}-\frac{5179}{104}a^{13}+\frac{1435}{104}a^{12}+\frac{20135}{52}a^{11}-\frac{5511}{52}a^{10}-\frac{35565}{26}a^{9}+\frac{733}{2}a^{8}+\frac{240865}{104}a^{7}-\frac{62161}{104}a^{6}-\frac{46721}{26}a^{5}+\frac{11303}{26}a^{4}+\frac{14649}{26}a^{3}-\frac{3297}{26}a^{2}-\frac{544}{13}a+\frac{142}{13}$, $a^{15}-a^{14}-\frac{1151}{52}a^{13}+\frac{1151}{52}a^{12}+\frac{8885}{52}a^{11}-\frac{8885}{52}a^{10}-\frac{1193}{2}a^{9}+\frac{1193}{2}a^{8}+\frac{12878}{13}a^{7}-\frac{12878}{13}a^{6}-\frac{38731}{52}a^{5}+\frac{38731}{52}a^{4}+\frac{2910}{13}a^{3}-\frac{2910}{13}a^{2}-\frac{214}{13}a+\frac{214}{13}$, $\frac{129}{104}a^{15}+\frac{227}{104}a^{14}-\frac{2837}{104}a^{13}-\frac{5049}{104}a^{12}+\frac{2705}{13}a^{11}+\frac{19655}{52}a^{10}-\frac{18483}{26}a^{9}-\frac{17403}{13}a^{8}+\frac{117913}{104}a^{7}+\frac{237027}{104}a^{6}-\frac{41039}{52}a^{5}-\frac{46591}{26}a^{4}+\frac{2758}{13}a^{3}+\frac{7557}{13}a^{2}-\frac{199}{13}a-\frac{636}{13}$, $\frac{9}{8}a^{15}+\frac{187}{104}a^{14}-\frac{2591}{104}a^{13}-\frac{4149}{104}a^{12}+\frac{10009}{52}a^{11}+\frac{16079}{52}a^{10}-673a^{9}-\frac{28251}{26}a^{8}+\frac{116645}{104}a^{7}+\frac{189755}{104}a^{6}-\frac{22179}{26}a^{5}-\frac{18221}{13}a^{4}+\frac{3507}{13}a^{3}+\frac{11513}{26}a^{2}-\frac{324}{13}a-\frac{492}{13}$, $\frac{147}{104}a^{15}-a^{14}-\frac{251}{8}a^{13}+\frac{1151}{52}a^{12}+\frac{3163}{13}a^{11}-\frac{8885}{52}a^{10}-\frac{22229}{26}a^{9}+\frac{1193}{2}a^{8}+\frac{148891}{104}a^{7}-\frac{12878}{13}a^{6}-\frac{56425}{52}a^{5}+\frac{38731}{52}a^{4}+\frac{4253}{13}a^{3}-\frac{2910}{13}a^{2}-\frac{328}{13}a+\frac{227}{13}$, $\frac{53}{104}a^{15}-\frac{19}{26}a^{14}-\frac{1189}{104}a^{13}+\frac{849}{52}a^{12}+\frac{1175}{13}a^{11}-\frac{6661}{52}a^{10}-\frac{4272}{13}a^{9}+\frac{11941}{26}a^{8}+\frac{60893}{104}a^{7}-\frac{20723}{26}a^{6}-\frac{25925}{52}a^{5}+\frac{33559}{52}a^{4}+\frac{4795}{26}a^{3}-\frac{2837}{13}a^{2}-\frac{293}{13}a+\frac{265}{13}$ (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: | \( 2097610.66881 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 2097610.66881 \cdot 1}{2\cdot\sqrt{176556947862350388527104}}\cr\approx \mathstrut & 0.163580831685 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2:C_8$ (as 16T24):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_2^2 : C_8$ |
Character table for $C_2^2 : C_8$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.4.157216.1, 4.4.9248.1, \(\Q(\zeta_{17})^+\), 8.8.420186801152.1, 8.8.24716870656.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 16 sibling: | deg 16 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
\(17\) | 17.16.14.1 | $x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |