Normalized defining polynomial
\( x^{16} - x^{15} - 11 x^{14} + 12 x^{13} + 43 x^{12} - 57 x^{11} - 68 x^{10} + 138 x^{9} + 26 x^{8} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[12, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(59904939404541015625\)
\(\medspace = 5^{12}\cdot 19^{2}\cdot 29^{4}\cdot 31^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.22\) | 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))
| |
Galois root discriminant: | $5^{3/4}19^{1/2}29^{1/2}31^{1/2}\approx 437.00272953194417$ | ||
Ramified primes: |
\(5\), \(19\), \(29\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{59}a^{15}+\frac{6}{59}a^{14}-\frac{28}{59}a^{13}-\frac{7}{59}a^{12}-\frac{6}{59}a^{11}+\frac{19}{59}a^{10}+\frac{6}{59}a^{9}+\frac{3}{59}a^{8}-\frac{12}{59}a^{7}+\frac{28}{59}a^{6}-\frac{11}{59}a^{5}+\frac{1}{59}a^{4}-\frac{13}{59}a^{3}-\frac{27}{59}a^{2}-\frac{12}{59}a-\frac{17}{59}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: |
$4a^{15}-4a^{14}-41a^{13}+47a^{12}+141a^{11}-214a^{10}-160a^{9}+471a^{8}-64a^{7}-504a^{6}+220a^{5}+255a^{4}-116a^{3}-61a^{2}+18a+5$, $\frac{32}{59}a^{15}-\frac{103}{59}a^{14}-\frac{306}{59}a^{13}+\frac{1133}{59}a^{12}+\frac{752}{59}a^{11}-\frac{4525}{59}a^{10}+\frac{1077}{59}a^{9}+\frac{7943}{59}a^{8}-\frac{7110}{59}a^{7}-\frac{5889}{59}a^{6}+\frac{9855}{59}a^{5}+\frac{1507}{59}a^{4}-\frac{5077}{59}a^{3}-\frac{97}{59}a^{2}+\frac{973}{59}a+\frac{46}{59}$, $a$, $\frac{207}{59}a^{15}-\frac{115}{59}a^{14}-\frac{2138}{59}a^{13}+\frac{1501}{59}a^{12}+\frac{7549}{59}a^{11}-\frac{7808}{59}a^{10}-\frac{9968}{59}a^{9}+\frac{19560}{59}a^{8}+\frac{1941}{59}a^{7}-\frac{22465}{59}a^{6}+\frac{4685}{59}a^{5}+\frac{10532}{59}a^{4}-\frac{3104}{59}a^{3}-\frac{1636}{59}a^{2}+\frac{702}{59}a+\frac{21}{59}$, $\frac{70}{59}a^{15}-\frac{52}{59}a^{14}-\frac{721}{59}a^{13}+\frac{631}{59}a^{12}+\frac{2530}{59}a^{11}-\frac{3036}{59}a^{10}-\frac{3238}{59}a^{9}+\frac{7172}{59}a^{8}+\frac{222}{59}a^{7}-\frac{8188}{59}a^{6}+\frac{2239}{59}a^{5}+\frac{4200}{59}a^{4}-\frac{1441}{59}a^{3}-\frac{710}{59}a^{2}+\frac{281}{59}a-\frac{69}{59}$, $\frac{391}{59}a^{15}-\frac{309}{59}a^{14}-\frac{4045}{59}a^{13}+\frac{3753}{59}a^{12}+\frac{14233}{59}a^{11}-\frac{17941}{59}a^{10}-\frac{17950}{59}a^{9}+\frac{41824}{59}a^{8}-\frac{208}{59}a^{7}-\frac{47049}{59}a^{6}+\frac{14166}{59}a^{5}+\frac{24168}{59}a^{4}-\frac{7974}{59}a^{3}-\frac{5188}{59}a^{2}+\frac{1444}{59}a+\frac{197}{59}$, $\frac{69}{59}a^{15}-\frac{294}{59}a^{14}-\frac{634}{59}a^{13}+\frac{3175}{59}a^{12}+\frac{1356}{59}a^{11}-\frac{12377}{59}a^{10}+\frac{3600}{59}a^{9}+\frac{21270}{59}a^{8}-\frac{19000}{59}a^{7}-\frac{16594}{59}a^{6}+\frac{26676}{59}a^{5}+\frac{7090}{59}a^{4}-\frac{14290}{59}a^{3}-\frac{2748}{59}a^{2}+\frac{2594}{59}a+\frac{479}{59}$, $\frac{115}{59}a^{15}-\frac{254}{59}a^{14}-\frac{1155}{59}a^{13}+\frac{2853}{59}a^{12}+\frac{3558}{59}a^{11}-\frac{11916}{59}a^{10}-\frac{962}{59}a^{9}+\frac{23001}{59}a^{8}-\frac{13475}{59}a^{7}-\frac{21147}{59}a^{6}+\frac{22512}{59}a^{5}+\frac{9496}{59}a^{4}-\frac{12646}{59}a^{3}-\frac{2338}{59}a^{2}+\frac{2337}{59}a+\frac{228}{59}$, $\frac{157}{59}a^{15}-\frac{120}{59}a^{14}-\frac{1623}{59}a^{13}+\frac{1438}{59}a^{12}+\frac{5725}{59}a^{11}-\frac{6811}{59}a^{10}-\frac{7436}{59}a^{9}+\frac{15870}{59}a^{8}+\frac{1007}{59}a^{7}-\frac{18142}{59}a^{6}+\frac{3701}{59}a^{5}+\frac{9892}{59}a^{4}-\frac{1805}{59}a^{3}-\frac{2410}{59}a^{2}+\frac{240}{59}a+\frac{163}{59}$, $\frac{130}{59}a^{15}-\frac{223}{59}a^{14}-\frac{1339}{59}a^{13}+\frac{2571}{59}a^{12}+\frac{4471}{59}a^{11}-\frac{11218}{59}a^{10}-\frac{3409}{59}a^{9}+\frac{23223}{59}a^{8}-\frac{9702}{59}a^{7}-\frac{23618}{59}a^{6}+\frac{19751}{59}a^{5}+\frac{11694}{59}a^{4}-\frac{12074}{59}a^{3}-\frac{2684}{59}a^{2}+\frac{2275}{59}a+\frac{209}{59}$, $\frac{516}{59}a^{15}-\frac{444}{59}a^{14}-\frac{5303}{59}a^{13}+\frac{5297}{59}a^{12}+\frac{18380}{59}a^{11}-\frac{24652}{59}a^{10}-\frac{21979}{59}a^{9}+\frac{55533}{59}a^{8}-\frac{3360}{59}a^{7}-\frac{60128}{59}a^{6}+\frac{20343}{59}a^{5}+\frac{29957}{59}a^{4}-\frac{9835}{59}a^{3}-\frac{6557}{59}a^{2}+\frac{1478}{59}a+\frac{373}{59}$, $\frac{206}{59}a^{15}-\frac{357}{59}a^{14}-\frac{2051}{59}a^{13}+\frac{3986}{59}a^{12}+\frac{6434}{59}a^{11}-\frac{16618}{59}a^{10}-\frac{3661}{59}a^{9}+\frac{32183}{59}a^{8}-\frac{15452}{59}a^{7}-\frac{29750}{59}a^{6}+\frac{25877}{59}a^{5}+\frac{13658}{59}a^{4}-\frac{13180}{59}a^{3}-\frac{3615}{59}a^{2}+\frac{2248}{59}a+\frac{333}{59}$, $\frac{133}{59}a^{15}-\frac{264}{59}a^{14}-\frac{1364}{59}a^{13}+\frac{2963}{59}a^{12}+\frac{4453}{59}a^{11}-\frac{12400}{59}a^{10}-\frac{2742}{59}a^{9}+\frac{24235}{59}a^{8}-\frac{11449}{59}a^{7}-\frac{23357}{59}a^{6}+\frac{20190}{59}a^{5}+\frac{11874}{59}a^{4}-\frac{10520}{59}a^{3}-\frac{3296}{59}a^{2}+\frac{1767}{59}a+\frac{335}{59}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 12873.243619 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 12873.243619 \cdot 1}{2\cdot\sqrt{59904939404541015625}}\cr\approx \mathstrut & 0.13447648863 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\wr D_4$ (as 16T1340):
A solvable group of order 2048 |
The 119 conjugacy class representatives for $C_2^2\wr D_4$ |
Character table for $C_2^2\wr D_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.4.725.1, 8.6.407359375.1, 8.8.309593125.1, 8.6.249671875.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 siblings: | data not computed |
Degree 32 siblings: | data not computed |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | R | R | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
\(19\)
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(29\)
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |