Normalized defining polynomial
\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} + 114 x^{12} - 138 x^{11} + 108 x^{10} + 10 x^{9} - 19 x^{8} + \cdots + 16 \)
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: | \(38182730939089616896\) \(\medspace = 2^{16}\cdot 17^{12}\) | 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: | \(16.74\) | 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: | $2\cdot 17^{3/4}\approx 16.74428805718538$ | ||
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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{5}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}$, $\frac{1}{52}a^{11}+\frac{1}{52}a^{10}-\frac{3}{52}a^{9}+\frac{3}{26}a^{8}+\frac{5}{52}a^{7}+\frac{7}{52}a^{6}-\frac{1}{4}a^{5}-\frac{1}{26}a^{4}-\frac{1}{26}a^{3}-\frac{5}{13}a^{2}-\frac{3}{13}a+\frac{4}{13}$, $\frac{1}{208}a^{12}-\frac{1}{104}a^{11}+\frac{7}{208}a^{10}-\frac{3}{26}a^{9}-\frac{1}{16}a^{8}-\frac{17}{104}a^{7}+\frac{5}{208}a^{6}+\frac{3}{26}a^{5}-\frac{3}{13}a^{4}-\frac{23}{52}a^{3}+\frac{3}{13}a^{2}+\frac{7}{26}$, $\frac{1}{208}a^{13}-\frac{1}{208}a^{11}-\frac{7}{104}a^{10}+\frac{3}{208}a^{9}+\frac{5}{52}a^{8}+\frac{21}{208}a^{7}+\frac{3}{104}a^{6}+\frac{7}{52}a^{4}-\frac{3}{26}a^{3}+\frac{9}{26}a^{2}-\frac{1}{2}a+\frac{3}{13}$, $\frac{1}{7904}a^{14}-\frac{7}{7904}a^{13}-\frac{9}{7904}a^{12}-\frac{7}{7904}a^{11}-\frac{23}{7904}a^{10}+\frac{291}{7904}a^{9}-\frac{423}{7904}a^{8}+\frac{155}{7904}a^{7}+\frac{813}{3952}a^{6}-\frac{53}{988}a^{5}-\frac{185}{1976}a^{4}-\frac{225}{988}a^{3}-\frac{253}{988}a^{2}+\frac{43}{988}a+\frac{9}{38}$, $\frac{1}{703456}a^{15}+\frac{37}{703456}a^{14}-\frac{1153}{703456}a^{13}-\frac{1581}{703456}a^{12}+\frac{1949}{703456}a^{11}-\frac{4103}{703456}a^{10}-\frac{24327}{703456}a^{9}-\frac{19407}{703456}a^{8}+\frac{27023}{351728}a^{7}+\frac{16047}{351728}a^{6}-\frac{8459}{175864}a^{5}+\frac{3662}{21983}a^{4}+\frac{9560}{21983}a^{3}-\frac{36245}{87932}a^{2}+\frac{4711}{43966}a-\frac{13263}{43966}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{11}{988} a^{14} - \frac{77}{988} a^{13} + \frac{543}{1976} a^{12} - \frac{157}{247} a^{11} + \frac{2097}{1976} a^{10} - \frac{1321}{988} a^{9} + \frac{2113}{1976} a^{8} - \frac{31}{494} a^{7} + \frac{641}{1976} a^{6} - \frac{672}{247} a^{5} + \frac{1575}{247} a^{4} - \frac{1834}{247} a^{3} + \frac{894}{247} a^{2} - \frac{118}{247} a - \frac{20}{247} \) (order $4$) | 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: | $\frac{2453}{351728}a^{15}-\frac{36795}{703456}a^{14}+\frac{126157}{703456}a^{13}-\frac{20151}{54112}a^{12}+\frac{375837}{703456}a^{11}-\frac{413237}{703456}a^{10}+\frac{283847}{703456}a^{9}+\frac{130419}{703456}a^{8}+\frac{9719}{37024}a^{7}-\frac{516897}{175864}a^{6}+\frac{12733}{3382}a^{5}-\frac{153449}{175864}a^{4}+\frac{33121}{43966}a^{3}-\frac{182425}{87932}a^{2}+\frac{128473}{87932}a+\frac{3816}{21983}$, $\frac{4377}{351728}a^{15}-\frac{61739}{703456}a^{14}+\frac{188541}{703456}a^{13}-\frac{311273}{703456}a^{12}+\frac{246865}{703456}a^{11}+\frac{82393}{703456}a^{10}-\frac{689553}{703456}a^{9}+\frac{1507945}{703456}a^{8}-\frac{361583}{703456}a^{7}-\frac{1806637}{351728}a^{6}+\frac{420105}{87932}a^{5}+\frac{777539}{175864}a^{4}-\frac{761999}{87932}a^{3}+\frac{388137}{87932}a^{2}-\frac{105315}{87932}a+\frac{9637}{43966}$, $\frac{10493}{351728}a^{15}-\frac{157395}{703456}a^{14}+\frac{548969}{703456}a^{13}-\frac{90857}{54112}a^{12}+\frac{1754797}{703456}a^{11}-\frac{1910579}{703456}a^{10}+\frac{1135379}{703456}a^{9}+\frac{985053}{703456}a^{8}-\frac{2481}{37024}a^{7}-\frac{3788059}{351728}a^{6}+\frac{115915}{6764}a^{5}-\frac{1724823}{175864}a^{4}+\frac{106895}{87932}a^{3}+\frac{73287}{87932}a^{2}-\frac{21055}{87932}a-\frac{53}{43966}$, $\frac{7685}{351728}a^{15}-\frac{115275}{703456}a^{14}+\frac{391225}{703456}a^{13}-\frac{61125}{54112}a^{12}+\frac{1072693}{703456}a^{11}-\frac{1005279}{703456}a^{10}+\frac{336251}{703456}a^{9}+\frac{1157313}{703456}a^{8}-\frac{3321}{37024}a^{7}-\frac{3137089}{351728}a^{6}+\frac{40189}{3382}a^{5}-\frac{463917}{175864}a^{4}-\frac{281943}{87932}a^{3}+\frac{142549}{87932}a^{2}+\frac{140321}{87932}a-\frac{38483}{43966}$, $\frac{7989}{175864}a^{15}-\frac{224985}{703456}a^{14}+\frac{725103}{703456}a^{13}-\frac{1432083}{703456}a^{12}+\frac{1964007}{703456}a^{11}-\frac{2014293}{703456}a^{10}+\frac{945653}{703456}a^{9}+\frac{129007}{54112}a^{8}+\frac{1136289}{703456}a^{7}-\frac{5730219}{351728}a^{6}+\frac{1484955}{87932}a^{5}-\frac{1001887}{175864}a^{4}+\frac{302717}{87932}a^{3}-\frac{47903}{87932}a^{2}-\frac{56219}{87932}a-\frac{35343}{43966}$, $\frac{10493}{351728}a^{15}-\frac{165227}{703456}a^{14}+\frac{603793}{703456}a^{13}-\frac{1374449}{703456}a^{12}+\frac{2201933}{703456}a^{11}-\frac{2657111}{703456}a^{10}+\frac{159687}{54112}a^{9}+\frac{232825}{703456}a^{8}-\frac{2995}{703456}a^{7}-\frac{3902157}{351728}a^{6}+\frac{1746127}{87932}a^{5}-\frac{2846223}{175864}a^{4}+\frac{759799}{87932}a^{3}-\frac{244977}{87932}a^{2}+\frac{20953}{87932}a+\frac{3507}{43966}$, $\frac{583}{27056}a^{15}-\frac{105853}{703456}a^{14}+\frac{372755}{703456}a^{13}-\frac{861733}{703456}a^{12}+\frac{1431491}{703456}a^{11}-\frac{1773755}{703456}a^{10}+\frac{1352761}{703456}a^{9}+\frac{54773}{703456}a^{8}+\frac{313835}{703456}a^{7}-\frac{895229}{175864}a^{6}+\frac{269300}{21983}a^{5}-\frac{2533321}{175864}a^{4}+\frac{117003}{21983}a^{3}+\frac{143831}{87932}a^{2}-\frac{5409}{87932}a-\frac{10441}{21983}$ | 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: | \( 2836.82177654 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2836.82177654 \cdot 2}{4\cdot\sqrt{38182730939089616896}}\cr\approx \mathstrut & 0.557580413623 \end{aligned}\]
Galois group
$C_2^2:C_4$ (as 16T10):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_2^2 : C_4$ |
Character table for $C_2^2 : C_4$ |
Intermediate fields
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 | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(17\) | 17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |