Normalized defining polynomial
\( x^{16} + 4x^{14} - 112x^{12} - 78x^{10} + 2650x^{8} - 164x^{6} - 16463x^{4} + 1206x^{2} + 4489 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(49535258684630680881135616\) \(\medspace = 2^{16}\cdot 17^{14}\cdot 67^{2}\) | 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: | \(40.36\) | 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: | $2^{15/8}17^{7/8}67^{1/2}\approx 358.1883442427048$ | ||
Ramified primes: | \(2\), \(17\), \(67\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{12}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{7}-\frac{1}{4}a^{5}$, $\frac{1}{200831607068756}a^{14}+\frac{3944143436229}{100415803534378}a^{12}+\frac{5643058023981}{50207901767189}a^{10}-\frac{4929426432534}{50207901767189}a^{8}+\frac{46875808853005}{200831607068756}a^{6}+\frac{18165833826973}{100415803534378}a^{4}+\frac{89998260436571}{200831607068756}a^{2}-\frac{605428162321}{2997486672668}$, $\frac{1}{200831607068756}a^{15}+\frac{3944143436229}{100415803534378}a^{13}+\frac{5643058023981}{50207901767189}a^{11}-\frac{4929426432534}{50207901767189}a^{9}+\frac{46875808853005}{200831607068756}a^{7}+\frac{18165833826973}{100415803534378}a^{5}+\frac{89998260436571}{200831607068756}a^{3}-\frac{605428162321}{2997486672668}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $\frac{7187074445}{200831607068756}a^{14}+\frac{5168734821}{50207901767189}a^{12}-\frac{205488569552}{50207901767189}a^{10}+\frac{659469717053}{200831607068756}a^{8}+\frac{19227492507499}{200831607068756}a^{6}-\frac{11878662787900}{50207901767189}a^{4}-\frac{36115139401055}{50207901767189}a^{2}+\frac{1257574404993}{749371668167}$, $\frac{5741598609}{200831607068756}a^{14}+\frac{24520647957}{100415803534378}a^{12}-\frac{709668171903}{200831607068756}a^{10}-\frac{4200939973935}{200831607068756}a^{8}+\frac{28179690497977}{200831607068756}a^{6}+\frac{76614297954947}{200831607068756}a^{4}-\frac{279719127070853}{200831607068756}a^{2}-\frac{236772901564}{749371668167}$, $\frac{3827449023}{100415803534378}a^{14}+\frac{44296987709}{100415803534378}a^{12}-\frac{110103966258}{50207901767189}a^{10}-\frac{5378649733213}{200831607068756}a^{8}+\frac{264128878602}{50207901767189}a^{6}+\frac{23589571023761}{100415803534378}a^{4}+\frac{106255502623289}{200831607068756}a^{2}+\frac{3293386335525}{2997486672668}$, $\frac{60185130165}{200831607068756}a^{14}+\frac{251389365253}{200831607068756}a^{12}-\frac{3279190372807}{100415803534378}a^{10}-\frac{5409480704751}{200831607068756}a^{8}+\frac{35764914011119}{50207901767189}a^{6}+\frac{11233444943353}{200831607068756}a^{4}-\frac{354967120337781}{100415803534378}a^{2}+\frac{12563258332}{749371668167}$, $\frac{13440462621}{100415803534378}a^{14}+\frac{115433532171}{100415803534378}a^{12}-\frac{1101659694619}{100415803534378}a^{10}-\frac{12816746153889}{200831607068756}a^{8}+\frac{9881313934795}{50207901767189}a^{6}+\frac{37422187440233}{50207901767189}a^{4}-\frac{154772111649437}{200831607068756}a^{2}-\frac{2171428130979}{2997486672668}$, $\frac{24175278213}{200831607068756}a^{14}+\frac{10865540211}{50207901767189}a^{12}-\frac{3025177250017}{200831607068756}a^{10}+\frac{3306672702271}{200831607068756}a^{8}+\frac{76953589006003}{200831607068756}a^{6}-\frac{101421461714617}{200831607068756}a^{4}-\frac{466582805095025}{200831607068756}a^{2}+\frac{1318998249133}{1498743336334}$, $\frac{52530232119}{200831607068756}a^{14}+\frac{162795389835}{200831607068756}a^{12}-\frac{3058982440291}{100415803534378}a^{10}-\frac{15415485769}{100415803534378}a^{8}+\frac{35500785132517}{50207901767189}a^{6}-\frac{35945697104169}{200831607068756}a^{4}-\frac{816189743298851}{200831607068756}a^{2}-\frac{6240619974865}{2997486672668}$, $\frac{305976734925}{100415803534378}a^{15}+\frac{337017132879}{200831607068756}a^{14}-\frac{1451870286705}{100415803534378}a^{13}+\frac{316773828589}{50207901767189}a^{12}+\frac{32979691875771}{100415803534378}a^{11}-\frac{9837327266543}{50207901767189}a^{10}+\frac{93328730630735}{200831607068756}a^{9}-\frac{7025782406102}{50207901767189}a^{8}-\frac{380283957038822}{50207901767189}a^{7}+\frac{996453705814025}{200831607068756}a^{6}-\frac{213510418273889}{50207901767189}a^{5}+\frac{73424676125339}{50207901767189}a^{4}+\frac{91\!\cdots\!27}{200831607068756}a^{3}-\frac{65\!\cdots\!03}{200831607068756}a^{2}+\frac{68332577881061}{2997486672668}a-\frac{53256898667851}{2997486672668}$, $\frac{175464200742}{50207901767189}a^{15}+\frac{20741194913}{2997486672668}a^{14}-\frac{5381777847879}{200831607068756}a^{13}+\frac{38739908781}{749371668167}a^{12}+\frac{59254797942925}{200831607068756}a^{11}-\frac{445168923750}{749371668167}a^{10}+\frac{274857349199559}{200831607068756}a^{9}-\frac{7766180965025}{2997486672668}a^{8}-\frac{886363362997919}{200831607068756}a^{7}+\frac{27644868387371}{2997486672668}a^{6}-\frac{818995038449492}{50207901767189}a^{5}+\frac{22814621854176}{749371668167}a^{4}+\frac{47831286226873}{100415803534378}a^{3}-\frac{9339963968023}{1498743336334}a^{2}+\frac{15002455819411}{2997486672668}a-\frac{13565347217347}{1498743336334}$, $\frac{5171047825}{50207901767189}a^{15}+\frac{375006499883}{200831607068756}a^{14}+\frac{1041829108627}{200831607068756}a^{13}+\frac{4032014440617}{200831607068756}a^{12}+\frac{12387336973627}{200831607068756}a^{11}-\frac{10103896646247}{100415803534378}a^{10}-\frac{77507731754981}{200831607068756}a^{9}-\frac{107809756083659}{100415803534378}a^{8}-\frac{561332687386913}{200831607068756}a^{7}-\frac{42447568593319}{100415803534378}a^{6}+\frac{174262296571626}{50207901767189}a^{5}+\frac{19\!\cdots\!01}{200831607068756}a^{4}+\frac{12\!\cdots\!57}{50207901767189}a^{3}+\frac{35\!\cdots\!25}{200831607068756}a^{2}+\frac{32039666686807}{2997486672668}a+\frac{18146303761791}{2997486672668}$, $\frac{499718534235}{100415803534378}a^{15}-\frac{185533706836}{50207901767189}a^{14}+\frac{2333261183889}{100415803534378}a^{13}-\frac{1789667498881}{100415803534378}a^{12}-\frac{27059464686315}{50207901767189}a^{11}+\frac{39529970993021}{100415803534378}a^{10}-\frac{35956972121986}{50207901767189}a^{9}+\frac{54623568539225}{100415803534378}a^{8}+\frac{633239439429288}{50207901767189}a^{7}-\frac{927802235590863}{100415803534378}a^{6}+\frac{659156722236667}{100415803534378}a^{5}-\frac{254284549328228}{50207901767189}a^{4}-\frac{77\!\cdots\!61}{100415803534378}a^{3}+\frac{28\!\cdots\!42}{50207901767189}a^{2}-\frac{56484642115729}{1498743336334}a+\frac{43930798035831}{1498743336334}$ (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: | \( 7477743.895676143 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 7477743.895676143 \cdot 1}{2\cdot\sqrt{49535258684630680881135616}}\cr\approx \mathstrut & 0.211954583406009 \end{aligned}\] (assuming GRH)
Galois group
$C_2^7:C_8$ (as 16T1194):
A solvable group of order 1024 |
The 40 conjugacy class representatives for $C_2^7:C_8$ |
Character table for $C_2^7:C_8$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
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 | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.10 | $x^{8} - 6 x^{7} + 36 x^{6} - 60 x^{5} + 88 x^{4} + 136 x^{3} + 336 x^{2} + 400 x + 144$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ |
2.8.8.10 | $x^{8} - 6 x^{7} + 36 x^{6} - 60 x^{5} + 88 x^{4} + 136 x^{3} + 336 x^{2} + 400 x + 144$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
\(17\) | 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
\(67\) | $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |