Normalized defining polynomial
\( x^{16} + 15x^{14} + 106x^{12} + 475x^{10} + 1486x^{8} + 3230x^{6} + 4576x^{4} + 3715x^{2} + 1296 \)
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: | \(540360087662636962890625\) \(\medspace = 5^{12}\cdot 19^{12}\) | 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: | \(30.43\) | 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: | $5^{3/4}19^{3/4}\approx 30.429351993705872$ | ||
Ramified primes: | \(5\), \(19\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{70}a^{12}+\frac{1}{10}a^{10}-\frac{8}{35}a^{8}+\frac{1}{70}a^{6}-\frac{1}{2}a^{5}-\frac{3}{10}a^{4}+\frac{1}{10}a^{2}+\frac{13}{35}$, $\frac{1}{70}a^{13}+\frac{1}{10}a^{11}-\frac{8}{35}a^{9}+\frac{1}{70}a^{7}-\frac{1}{2}a^{6}-\frac{3}{10}a^{5}+\frac{1}{10}a^{3}+\frac{13}{35}a$, $\frac{1}{4130}a^{14}+\frac{9}{4130}a^{12}-\frac{597}{4130}a^{10}-\frac{243}{2065}a^{8}-\frac{867}{2065}a^{6}+\frac{17}{118}a^{4}-\frac{1}{2}a^{3}-\frac{377}{826}a^{2}-\frac{954}{2065}$, $\frac{1}{148680}a^{15}-\frac{23}{9912}a^{13}-\frac{101}{14868}a^{11}+\frac{7243}{148680}a^{9}-\frac{3109}{74340}a^{7}-\frac{2819}{10620}a^{5}+\frac{10783}{37170}a^{3}-\frac{1}{2}a^{2}+\frac{28123}{148680}a-\frac{1}{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{2}\times C_{8}$, which has order $16$ (assuming GRH)
Unit group
Rank: | $7$ | 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: | $\frac{59}{2520}a^{15}+\frac{283}{840}a^{13}+\frac{2839}{1260}a^{11}+\frac{23813}{2520}a^{9}+\frac{34441}{1260}a^{7}+\frac{9503}{180}a^{5}+\frac{18817}{315}a^{3}+\frac{72341}{2520}a+\frac{1}{2}$, $\frac{2263}{148680}a^{15}+\frac{149}{4130}a^{14}+\frac{10211}{49560}a^{13}+\frac{199}{413}a^{12}+\frac{98663}{74340}a^{11}+\frac{2529}{826}a^{10}+\frac{115303}{21240}a^{9}+\frac{51427}{4130}a^{8}+\frac{1138547}{74340}a^{7}+\frac{72479}{2065}a^{6}+\frac{304831}{10620}a^{5}+\frac{38743}{590}a^{4}+\frac{579764}{18585}a^{3}+\frac{150939}{2065}a^{2}+\frac{1997497}{148680}a+\frac{147647}{4130}$, $\frac{1}{10620}a^{15}+\frac{611}{24780}a^{13}+\frac{1619}{5310}a^{11}+\frac{131413}{74340}a^{9}+\frac{240551}{37170}a^{7}+\frac{17081}{1062}a^{5}+\frac{12989}{531}a^{3}+\frac{1273729}{74340}a$, $\frac{151}{18585}a^{15}+\frac{43}{2065}a^{14}+\frac{689}{6195}a^{13}+\frac{564}{2065}a^{12}+\frac{25793}{37170}a^{11}+\frac{6891}{4130}a^{10}+\frac{14441}{5310}a^{9}+\frac{384}{59}a^{8}+\frac{137419}{18585}a^{7}+\frac{14437}{826}a^{6}+\frac{34901}{2655}a^{5}+\frac{17753}{590}a^{4}+\frac{238636}{18585}a^{3}+\frac{119143}{4130}a^{2}+\frac{19574}{3717}a+\frac{23743}{2065}$, $\frac{17}{2360}a^{15}-\frac{5}{59}a^{14}+\frac{2369}{16520}a^{13}-\frac{925}{826}a^{12}+\frac{1327}{1180}a^{11}-\frac{815}{118}a^{10}+\frac{84533}{16520}a^{9}-\frac{11192}{413}a^{8}+\frac{130821}{8260}a^{7}-\frac{60635}{826}a^{6}+\frac{38953}{1180}a^{5}-\frac{15213}{118}a^{4}+\frac{23067}{590}a^{3}-\frac{7508}{59}a^{2}+\frac{329141}{16520}a-\frac{43027}{826}$, $\frac{431}{24780}a^{15}+\frac{82}{2065}a^{14}+\frac{2119}{8260}a^{13}+\frac{2361}{4130}a^{12}+\frac{10736}{6195}a^{11}+\frac{15667}{4130}a^{10}+\frac{182471}{24780}a^{9}+\frac{65141}{4130}a^{8}+\frac{133856}{6195}a^{7}+\frac{187329}{4130}a^{6}+\frac{37759}{885}a^{5}+\frac{5111}{59}a^{4}+\frac{625663}{12390}a^{3}+\frac{79831}{826}a^{2}+\frac{132973}{4956}a+\frac{92524}{2065}$, $\frac{431}{24780}a^{15}-\frac{82}{2065}a^{14}+\frac{2119}{8260}a^{13}-\frac{2361}{4130}a^{12}+\frac{10736}{6195}a^{11}-\frac{15667}{4130}a^{10}+\frac{182471}{24780}a^{9}-\frac{65141}{4130}a^{8}+\frac{133856}{6195}a^{7}-\frac{187329}{4130}a^{6}+\frac{37759}{885}a^{5}-\frac{5111}{59}a^{4}+\frac{625663}{12390}a^{3}-\frac{79831}{826}a^{2}+\frac{132973}{4956}a-\frac{92524}{2065}$ (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: | \( 71285.5480074 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 71285.5480074 \cdot 16}{2\cdot\sqrt{540360087662636962890625}}\cr\approx \mathstrut & 1.88446811969 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3:C_4$ (as 16T33):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3:C_4$ |
Character table for $C_2^3:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-95}) \), 4.0.45125.1 x2, \(\Q(\sqrt{5}, \sqrt{-19})\), 4.2.2375.1 x2, 8.0.735091890625.1 x2, 8.0.2036265625.2, 8.0.147018378125.1 x2 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(19\) | 19.8.6.2 | $x^{8} + 72 x^{7} + 1952 x^{6} + 23760 x^{5} + 112814 x^{4} + 48888 x^{3} + 44288 x^{2} + 435600 x + 1945825$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |
19.8.6.2 | $x^{8} + 72 x^{7} + 1952 x^{6} + 23760 x^{5} + 112814 x^{4} + 48888 x^{3} + 44288 x^{2} + 435600 x + 1945825$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |