Normalized defining polynomial
\( x^{16} - 11x^{14} + 96x^{12} - 781x^{10} + 6191x^{8} - 19525x^{6} + 60000x^{4} - 171875x^{2} + 390625 \)
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$$\chi_{420}(1,·)$, $\chi_{420}(209,·)$, $\chi_{420}(337,·)$, $\chi_{420}(83,·)$, $\chi_{420}(251,·)$, $\chi_{420}(419,·)$, $\chi_{420}(293,·)$, $\chi_{420}(167,·)$, $\chi_{420}(41,·)$, $\chi_{420}(43,·)$, $\chi_{420}(211,·)$, $\chi_{420}(169,·)$, $\chi_{420}(377,·)$, $\chi_{420}(379,·)$, $\chi_{420}(253,·)$, $\chi_{420}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{7}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{154775}a^{10}+\frac{4}{25}a^{8}+\frac{6}{25}a^{6}+\frac{9}{25}a^{4}+\frac{1}{25}a^{2}-\frac{781}{6191}$, $\frac{1}{773875}a^{11}+\frac{4}{125}a^{9}+\frac{56}{125}a^{7}+\frac{34}{125}a^{5}-\frac{24}{125}a^{3}+\frac{1082}{6191}a$, $\frac{1}{3869375}a^{12}-\frac{11}{3869375}a^{10}-\frac{44}{625}a^{8}-\frac{241}{625}a^{6}+\frac{1}{625}a^{4}-\frac{781}{154775}a^{2}+\frac{96}{6191}$, $\frac{1}{19346875}a^{13}-\frac{11}{19346875}a^{11}-\frac{44}{3125}a^{9}+\frac{1009}{3125}a^{7}-\frac{624}{3125}a^{5}+\frac{153994}{773875}a^{3}-\frac{1219}{6191}a$, $\frac{1}{96734375}a^{14}-\frac{11}{96734375}a^{12}+\frac{96}{96734375}a^{10}-\frac{2741}{15625}a^{8}+\frac{1}{15625}a^{6}-\frac{781}{3869375}a^{4}+\frac{96}{154775}a^{2}-\frac{11}{6191}$, $\frac{1}{483671875}a^{15}-\frac{11}{483671875}a^{13}+\frac{96}{483671875}a^{11}-\frac{2741}{78125}a^{9}+\frac{15626}{78125}a^{7}-\frac{3870156}{19346875}a^{5}+\frac{154871}{773875}a^{3}-\frac{6202}{30955}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}\times C_{8}$, which has order $32$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{3311}{483671875} a^{15} + \frac{28896}{483671875} a^{13} - \frac{235081}{483671875} a^{11} + \frac{301}{78125} a^{9} - \frac{2286}{78125} a^{7} + \frac{28896}{773875} a^{5} - \frac{3311}{30955} a^{3} + \frac{1505}{6191} a \) (order $20$) | 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: | $\frac{1}{154775}a^{12}+\frac{48576}{154775}a^{2}-1$, $\frac{1}{30955}a^{11}+\frac{17621}{30955}a+1$, $\frac{2286}{483671875}a^{15}-\frac{25146}{483671875}a^{13}+\frac{1}{154775}a^{12}+\frac{219456}{483671875}a^{11}-\frac{301}{78125}a^{9}+\frac{2286}{78125}a^{7}-\frac{1785366}{19346875}a^{5}+\frac{219456}{773875}a^{3}+\frac{48576}{154775}a^{2}-\frac{25146}{30955}a$, $\frac{27}{19346875}a^{15}+\frac{228127}{19346875}a^{5}$, $\frac{6077}{483671875}a^{15}-\frac{376}{96734375}a^{14}-\frac{89497}{483671875}a^{13}-\frac{2339}{96734375}a^{12}+\frac{860667}{483671875}a^{11}+\frac{69504}{96734375}a^{10}-\frac{1082}{78125}a^{9}-\frac{109}{15625}a^{8}+\frac{8477}{78125}a^{7}+\frac{724}{15625}a^{6}-\frac{9581308}{19346875}a^{5}-\frac{2083688}{3869375}a^{4}+\frac{950473}{773875}a^{3}+\frac{351913}{154775}a^{2}-\frac{40384}{30955}a-\frac{26537}{6191}$, $\frac{29401}{483671875}a^{15}-\frac{9271}{96734375}a^{14}-\frac{251161}{483671875}a^{13}+\frac{89656}{96734375}a^{12}+\frac{2082121}{483671875}a^{11}-\frac{738816}{96734375}a^{10}-\frac{2766}{78125}a^{9}+\frac{986}{15625}a^{8}+\frac{21526}{78125}a^{7}-\frac{7696}{15625}a^{6}-\frac{7701366}{19346875}a^{5}+\frac{4188488}{3869375}a^{4}+\frac{1413573}{773875}a^{3}-\frac{504983}{154775}a^{2}-\frac{43833}{6191}a+\frac{84656}{6191}$, $\frac{1334}{483671875}a^{15}+\frac{257}{96734375}a^{14}-\frac{23424}{483671875}a^{13}+\frac{3748}{96734375}a^{12}+\frac{96814}{483671875}a^{11}+\frac{36722}{96734375}a^{10}-\frac{244}{78125}a^{9}+\frac{13}{15625}a^{8}+\frac{1334}{78125}a^{7}+\frac{57}{15625}a^{6}-\frac{1041854}{19346875}a^{5}-\frac{120234}{3869375}a^{4}+\frac{2684}{30955}a^{3}+\frac{35954}{154775}a^{2}-\frac{18961}{30955}a+\frac{3660}{6191}$ (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: | \( 107520.786508 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 107520.786508 \cdot 32}{20\cdot\sqrt{605165749776000000000000}}\cr\approx \mathstrut & 0.537173020129 \end{aligned}\] (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 | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.1.0.1}{1} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
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 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(7\) | 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 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |