Normalized defining polynomial
\( x^{17} - x^{16} - x^{15} - x^{14} + x^{12} + 13 x^{11} + 7 x^{10} + 11 x^{9} + 4 x^{8} + x^{7} + \cdots - 1 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(463009808974713123841\) \(\medspace = 383^{8}\) | 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: | \(16.43\) | 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: | $383^{1/2}\approx 19.570385790780925$ | ||
Ramified primes: | \(383\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{65}a^{14}+\frac{11}{65}a^{13}-\frac{3}{13}a^{12}-\frac{31}{65}a^{11}+\frac{9}{65}a^{10}+\frac{1}{13}a^{9}+\frac{28}{65}a^{8}+\frac{12}{65}a^{7}-\frac{28}{65}a^{6}+\frac{24}{65}a^{5}-\frac{31}{65}a^{4}+\frac{3}{65}a^{3}-\frac{6}{65}a^{2}+\frac{2}{13}a+\frac{1}{65}$, $\frac{1}{715}a^{15}-\frac{2}{715}a^{14}-\frac{93}{715}a^{13}-\frac{226}{715}a^{12}-\frac{173}{715}a^{11}-\frac{307}{715}a^{10}+\frac{93}{715}a^{9}-\frac{92}{715}a^{8}+\frac{271}{715}a^{7}-\frac{67}{715}a^{6}-\frac{18}{715}a^{5}-\frac{244}{715}a^{4}-\frac{35}{143}a^{3}+\frac{218}{715}a^{2}-\frac{259}{715}a-\frac{1}{55}$, $\frac{1}{3754465}a^{16}+\frac{382}{750893}a^{15}-\frac{4687}{3754465}a^{14}+\frac{1760433}{3754465}a^{13}-\frac{328105}{750893}a^{12}+\frac{1358022}{3754465}a^{11}+\frac{1837179}{3754465}a^{10}+\frac{466309}{3754465}a^{9}+\frac{1345062}{3754465}a^{8}-\frac{301062}{750893}a^{7}-\frac{4218}{42185}a^{6}-\frac{205108}{750893}a^{5}+\frac{1173782}{3754465}a^{4}-\frac{654867}{3754465}a^{3}-\frac{940898}{3754465}a^{2}+\frac{1634929}{3754465}a+\frac{1364334}{3754465}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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)
| |
Fundamental units: | $\frac{17794}{750893}a^{16}+\frac{199376}{3754465}a^{15}-\frac{119618}{750893}a^{14}-\frac{59041}{3754465}a^{13}-\frac{9616}{3754465}a^{12}-\frac{1695}{750893}a^{11}+\frac{955431}{3754465}a^{10}+\frac{4732503}{3754465}a^{9}+\frac{185949}{3754465}a^{8}+\frac{63138}{68263}a^{7}+\frac{9793}{42185}a^{6}+\frac{168257}{750893}a^{5}+\frac{3140734}{3754465}a^{4}+\frac{9329486}{3754465}a^{3}+\frac{698291}{341315}a^{2}+\frac{11270456}{3754465}a+\frac{3698674}{3754465}$, $\frac{59439}{750893}a^{16}-\frac{573511}{3754465}a^{15}+\frac{168841}{3754465}a^{14}-\frac{100703}{3754465}a^{13}-\frac{183319}{3754465}a^{12}+\frac{13934}{341315}a^{11}+\frac{3391528}{3754465}a^{10}-\frac{675933}{3754465}a^{9}+\frac{4026569}{3754465}a^{8}+\frac{899942}{3754465}a^{7}-\frac{2229}{8437}a^{6}+\frac{2313189}{3754465}a^{5}+\frac{334168}{750893}a^{4}+\frac{5705047}{3754465}a^{3}+\frac{9056613}{3754465}a^{2}+\frac{376484}{341315}a+\frac{1693162}{3754465}$, $\frac{109667}{750893}a^{16}-\frac{1084308}{3754465}a^{15}+\frac{575654}{3754465}a^{14}-\frac{831093}{3754465}a^{13}+\frac{7523}{341315}a^{12}+\frac{387421}{3754465}a^{11}+\frac{620053}{341315}a^{10}-\frac{186458}{288805}a^{9}+\frac{1886628}{750893}a^{8}-\frac{3178152}{3754465}a^{7}+\frac{25748}{42185}a^{6}+\frac{443151}{341315}a^{5}+\frac{770209}{341315}a^{4}+\frac{9093189}{3754465}a^{3}+\frac{18983938}{3754465}a^{2}+\frac{145242}{3754465}a+\frac{2439452}{3754465}$, $\frac{271369}{3754465}a^{16}-\frac{704552}{3754465}a^{15}+\frac{353036}{3754465}a^{14}-\frac{184037}{3754465}a^{13}+\frac{300827}{3754465}a^{12}+\frac{655609}{3754465}a^{11}+\frac{45406}{57761}a^{10}-\frac{824275}{750893}a^{9}+\frac{2424222}{3754465}a^{8}-\frac{3331872}{3754465}a^{7}-\frac{1286}{3835}a^{6}+\frac{2059516}{3754465}a^{5}+\frac{2140761}{3754465}a^{4}+\frac{434102}{3754465}a^{3}+\frac{291822}{3754465}a^{2}-\frac{10532001}{3754465}a-\frac{2936398}{3754465}$, $\frac{1512}{68263}a^{16}+\frac{82117}{3754465}a^{15}-\frac{351509}{3754465}a^{14}+\frac{116854}{3754465}a^{13}-\frac{599897}{3754465}a^{12}+\frac{109819}{3754465}a^{11}+\frac{1414536}{3754465}a^{10}+\frac{270187}{288805}a^{9}+\frac{186981}{3754465}a^{8}+\frac{3189827}{3754465}a^{7}-\frac{12641}{42185}a^{6}+\frac{1780149}{3754465}a^{5}+\frac{1276447}{3754465}a^{4}+\frac{1567078}{750893}a^{3}+\frac{9924196}{3754465}a^{2}+\frac{8036577}{3754465}a+\frac{1254744}{3754465}$, $\frac{27813}{341315}a^{16}-\frac{69417}{3754465}a^{15}-\frac{683889}{3754465}a^{14}-\frac{375197}{3754465}a^{13}-\frac{8923}{3754465}a^{12}+\frac{157059}{3754465}a^{11}+\frac{4389928}{3754465}a^{10}+\frac{4669861}{3754465}a^{9}+\frac{3395244}{3754465}a^{8}+\frac{262353}{288805}a^{7}+\frac{20841}{42185}a^{6}+\frac{931023}{3754465}a^{5}+\frac{9776836}{3754465}a^{4}+\frac{11871193}{3754465}a^{3}+\frac{14342672}{3754465}a^{2}+\frac{2260768}{750893}a+\frac{2083381}{3754465}$, $\frac{665628}{3754465}a^{16}-\frac{647509}{3754465}a^{15}-\frac{181845}{750893}a^{14}+\frac{54429}{3754465}a^{13}-\frac{65216}{341315}a^{12}+\frac{136134}{750893}a^{11}+\frac{800827}{341315}a^{10}+\frac{1044944}{750893}a^{9}+\frac{4911183}{3754465}a^{8}+\frac{6927207}{3754465}a^{7}-\frac{43783}{42185}a^{6}+\frac{543424}{341315}a^{5}+\frac{100498}{26255}a^{4}+\frac{20485768}{3754465}a^{3}+\frac{27202731}{3754465}a^{2}+\frac{16622978}{3754465}a-\frac{586758}{3754465}$, $\frac{762276}{3754465}a^{16}-\frac{401937}{3754465}a^{15}-\frac{1267452}{3754465}a^{14}-\frac{201466}{750893}a^{13}-\frac{22098}{341315}a^{12}+\frac{1053487}{3754465}a^{11}+\frac{959702}{341315}a^{10}+\frac{710981}{288805}a^{9}+\frac{8996164}{3754465}a^{8}+\frac{1131855}{750893}a^{7}+\frac{10146}{42185}a^{6}+\frac{6921}{5251}a^{5}+\frac{1857559}{341315}a^{4}+\frac{28912051}{3754465}a^{3}+\frac{8355836}{750893}a^{2}+\frac{26267647}{3754465}a+\frac{9066576}{3754465}$ | 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: | \( 2610.31631075 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 2610.31631075 \cdot 1}{2\cdot\sqrt{463009808974713123841}}\cr\approx \mathstrut & 0.294670722603 \end{aligned}\]
Galois group
A solvable group of order 34 |
The 10 conjugacy class representatives for $D_{17}$ |
Character table for $D_{17}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | 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 | $17$ | $17$ | ${\href{/padicField/5.2.0.1}{2} }^{8}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $17$ | ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $17$ | $17$ | $17$ | $17$ | $17$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17$ | ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(383\) | $\Q_{383}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |