Normalized defining polynomial
\( x^{17} - 2 x^{16} - 3 x^{14} + 24 x^{13} + 18 x^{12} - 138 x^{10} - 203 x^{9} - 75 x^{8} + 352 x^{7} + 588 x^{6} + 384 x^{5} + 132 x^{4} + 493 x^{3} + 1015 x^{2} + 874 x + 337 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(34895459505131153638432321\) \(\medspace = 1559^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(31.81\) | 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: | $1559^{1/2}\approx 39.48417404479927$ | ||
Ramified primes: | \(1559\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{443}a^{15}-\frac{22}{443}a^{14}+\frac{5}{443}a^{13}+\frac{164}{443}a^{12}-\frac{115}{443}a^{11}+\frac{86}{443}a^{10}+\frac{18}{443}a^{9}+\frac{190}{443}a^{8}+\frac{128}{443}a^{7}+\frac{214}{443}a^{6}+\frac{197}{443}a^{5}+\frac{132}{443}a^{4}+\frac{206}{443}a^{3}+\frac{169}{443}a^{2}+\frac{90}{443}a+\frac{124}{443}$, $\frac{1}{62\!\cdots\!87}a^{16}-\frac{59\!\cdots\!16}{62\!\cdots\!87}a^{15}+\frac{25\!\cdots\!83}{62\!\cdots\!87}a^{14}-\frac{19\!\cdots\!64}{62\!\cdots\!87}a^{13}-\frac{16\!\cdots\!89}{62\!\cdots\!87}a^{12}-\frac{27\!\cdots\!86}{62\!\cdots\!87}a^{11}-\frac{47\!\cdots\!68}{62\!\cdots\!87}a^{10}+\frac{32\!\cdots\!68}{62\!\cdots\!87}a^{9}+\frac{18\!\cdots\!28}{62\!\cdots\!87}a^{8}+\frac{66\!\cdots\!42}{62\!\cdots\!87}a^{7}-\frac{16\!\cdots\!79}{62\!\cdots\!87}a^{6}-\frac{13\!\cdots\!77}{60\!\cdots\!29}a^{5}+\frac{14\!\cdots\!15}{62\!\cdots\!87}a^{4}-\frac{39\!\cdots\!71}{62\!\cdots\!87}a^{3}-\frac{21\!\cdots\!39}{62\!\cdots\!87}a^{2}+\frac{30\!\cdots\!28}{60\!\cdots\!29}a+\frac{32\!\cdots\!80}{62\!\cdots\!87}$
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)
| |
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: | $\frac{80\!\cdots\!66}{62\!\cdots\!87}a^{16}-\frac{20\!\cdots\!22}{62\!\cdots\!87}a^{15}+\frac{59\!\cdots\!19}{62\!\cdots\!87}a^{14}-\frac{58\!\cdots\!47}{62\!\cdots\!87}a^{13}+\frac{16\!\cdots\!58}{62\!\cdots\!87}a^{12}+\frac{74\!\cdots\!47}{62\!\cdots\!87}a^{11}-\frac{16\!\cdots\!78}{62\!\cdots\!87}a^{10}-\frac{92\!\cdots\!04}{62\!\cdots\!87}a^{9}-\frac{10\!\cdots\!95}{62\!\cdots\!87}a^{8}+\frac{44\!\cdots\!52}{62\!\cdots\!87}a^{7}+\frac{25\!\cdots\!94}{62\!\cdots\!87}a^{6}+\frac{24\!\cdots\!23}{60\!\cdots\!29}a^{5}+\frac{49\!\cdots\!76}{62\!\cdots\!87}a^{4}+\frac{85\!\cdots\!48}{62\!\cdots\!87}a^{3}+\frac{46\!\cdots\!42}{62\!\cdots\!87}a^{2}+\frac{55\!\cdots\!67}{60\!\cdots\!29}a+\frac{24\!\cdots\!49}{62\!\cdots\!87}$, $\frac{10\!\cdots\!18}{62\!\cdots\!87}a^{16}-\frac{20\!\cdots\!62}{62\!\cdots\!87}a^{15}-\frac{43\!\cdots\!15}{62\!\cdots\!87}a^{14}-\frac{18\!\cdots\!89}{62\!\cdots\!87}a^{13}+\frac{25\!\cdots\!28}{62\!\cdots\!87}a^{12}+\frac{14\!\cdots\!87}{62\!\cdots\!87}a^{11}-\frac{86\!\cdots\!05}{62\!\cdots\!87}a^{10}-\frac{14\!\cdots\!81}{62\!\cdots\!87}a^{9}-\frac{17\!\cdots\!85}{62\!\cdots\!87}a^{8}-\frac{20\!\cdots\!21}{62\!\cdots\!87}a^{7}+\frac{43\!\cdots\!30}{62\!\cdots\!87}a^{6}+\frac{47\!\cdots\!10}{60\!\cdots\!29}a^{5}+\frac{20\!\cdots\!94}{62\!\cdots\!87}a^{4}-\frac{23\!\cdots\!44}{62\!\cdots\!87}a^{3}+\frac{57\!\cdots\!31}{62\!\cdots\!87}a^{2}+\frac{86\!\cdots\!46}{60\!\cdots\!29}a+\frac{56\!\cdots\!30}{62\!\cdots\!87}$, $\frac{42\!\cdots\!26}{62\!\cdots\!87}a^{16}-\frac{13\!\cdots\!32}{62\!\cdots\!87}a^{15}+\frac{16\!\cdots\!90}{62\!\cdots\!87}a^{14}-\frac{23\!\cdots\!59}{62\!\cdots\!87}a^{13}+\frac{11\!\cdots\!90}{62\!\cdots\!87}a^{12}-\frac{63\!\cdots\!43}{62\!\cdots\!87}a^{11}+\frac{73\!\cdots\!21}{62\!\cdots\!87}a^{10}-\frac{40\!\cdots\!45}{62\!\cdots\!87}a^{9}-\frac{18\!\cdots\!07}{62\!\cdots\!87}a^{8}+\frac{22\!\cdots\!65}{62\!\cdots\!87}a^{7}+\frac{36\!\cdots\!18}{62\!\cdots\!87}a^{6}+\frac{16\!\cdots\!19}{60\!\cdots\!29}a^{5}-\frac{27\!\cdots\!79}{62\!\cdots\!87}a^{4}+\frac{24\!\cdots\!19}{62\!\cdots\!87}a^{3}+\frac{46\!\cdots\!35}{62\!\cdots\!87}a^{2}+\frac{41\!\cdots\!68}{60\!\cdots\!29}a+\frac{12\!\cdots\!90}{62\!\cdots\!87}$, $\frac{90\!\cdots\!41}{62\!\cdots\!87}a^{16}-\frac{28\!\cdots\!63}{62\!\cdots\!87}a^{15}+\frac{29\!\cdots\!46}{62\!\cdots\!87}a^{14}-\frac{46\!\cdots\!39}{62\!\cdots\!87}a^{13}+\frac{25\!\cdots\!70}{62\!\cdots\!87}a^{12}-\frac{11\!\cdots\!21}{62\!\cdots\!87}a^{11}+\frac{13\!\cdots\!10}{62\!\cdots\!87}a^{10}-\frac{11\!\cdots\!12}{62\!\cdots\!87}a^{9}-\frac{43\!\cdots\!76}{62\!\cdots\!87}a^{8}+\frac{34\!\cdots\!27}{62\!\cdots\!87}a^{7}+\frac{28\!\cdots\!81}{62\!\cdots\!87}a^{6}+\frac{15\!\cdots\!98}{60\!\cdots\!29}a^{5}+\frac{39\!\cdots\!56}{62\!\cdots\!87}a^{4}+\frac{23\!\cdots\!55}{62\!\cdots\!87}a^{3}+\frac{43\!\cdots\!03}{62\!\cdots\!87}a^{2}+\frac{37\!\cdots\!08}{60\!\cdots\!29}a+\frac{17\!\cdots\!86}{62\!\cdots\!87}$, $\frac{13\!\cdots\!91}{62\!\cdots\!87}a^{16}-\frac{36\!\cdots\!18}{62\!\cdots\!87}a^{15}+\frac{85\!\cdots\!92}{62\!\cdots\!87}a^{14}+\frac{14\!\cdots\!89}{62\!\cdots\!87}a^{13}+\frac{25\!\cdots\!07}{62\!\cdots\!87}a^{12}+\frac{72\!\cdots\!80}{62\!\cdots\!87}a^{11}-\frac{36\!\cdots\!72}{62\!\cdots\!87}a^{10}-\frac{14\!\cdots\!54}{62\!\cdots\!87}a^{9}-\frac{12\!\cdots\!60}{62\!\cdots\!87}a^{8}+\frac{12\!\cdots\!49}{62\!\cdots\!87}a^{7}+\frac{43\!\cdots\!07}{62\!\cdots\!87}a^{6}+\frac{24\!\cdots\!33}{60\!\cdots\!29}a^{5}-\frac{22\!\cdots\!34}{62\!\cdots\!87}a^{4}+\frac{18\!\cdots\!26}{62\!\cdots\!87}a^{3}+\frac{61\!\cdots\!92}{62\!\cdots\!87}a^{2}+\frac{54\!\cdots\!76}{60\!\cdots\!29}a+\frac{23\!\cdots\!11}{62\!\cdots\!87}$, $\frac{16\!\cdots\!96}{62\!\cdots\!87}a^{16}-\frac{15\!\cdots\!27}{62\!\cdots\!87}a^{15}+\frac{43\!\cdots\!76}{62\!\cdots\!87}a^{14}-\frac{62\!\cdots\!59}{62\!\cdots\!87}a^{13}+\frac{12\!\cdots\!36}{62\!\cdots\!87}a^{12}-\frac{32\!\cdots\!22}{62\!\cdots\!87}a^{11}+\frac{23\!\cdots\!94}{62\!\cdots\!87}a^{10}-\frac{24\!\cdots\!41}{62\!\cdots\!87}a^{9}+\frac{10\!\cdots\!35}{62\!\cdots\!87}a^{8}+\frac{30\!\cdots\!08}{62\!\cdots\!87}a^{7}-\frac{49\!\cdots\!31}{62\!\cdots\!87}a^{6}-\frac{15\!\cdots\!97}{60\!\cdots\!29}a^{5}-\frac{15\!\cdots\!45}{62\!\cdots\!87}a^{4}+\frac{11\!\cdots\!98}{62\!\cdots\!87}a^{3}-\frac{13\!\cdots\!21}{62\!\cdots\!87}a^{2}-\frac{25\!\cdots\!19}{60\!\cdots\!29}a-\frac{30\!\cdots\!93}{62\!\cdots\!87}$, $\frac{36\!\cdots\!57}{62\!\cdots\!87}a^{16}-\frac{16\!\cdots\!58}{62\!\cdots\!87}a^{15}+\frac{79\!\cdots\!06}{62\!\cdots\!87}a^{14}+\frac{60\!\cdots\!44}{62\!\cdots\!87}a^{13}-\frac{45\!\cdots\!78}{62\!\cdots\!87}a^{12}-\frac{46\!\cdots\!98}{62\!\cdots\!87}a^{11}-\frac{26\!\cdots\!59}{62\!\cdots\!87}a^{10}-\frac{74\!\cdots\!29}{62\!\cdots\!87}a^{9}+\frac{68\!\cdots\!69}{62\!\cdots\!87}a^{8}+\frac{13\!\cdots\!71}{62\!\cdots\!87}a^{7}+\frac{20\!\cdots\!50}{62\!\cdots\!87}a^{6}-\frac{36\!\cdots\!02}{60\!\cdots\!29}a^{5}-\frac{86\!\cdots\!72}{62\!\cdots\!87}a^{4}+\frac{34\!\cdots\!07}{62\!\cdots\!87}a^{3}+\frac{49\!\cdots\!60}{62\!\cdots\!87}a^{2}-\frac{48\!\cdots\!20}{60\!\cdots\!29}a-\frac{19\!\cdots\!30}{62\!\cdots\!87}$, $\frac{25\!\cdots\!56}{62\!\cdots\!87}a^{16}-\frac{59\!\cdots\!71}{62\!\cdots\!87}a^{15}+\frac{15\!\cdots\!78}{62\!\cdots\!87}a^{14}-\frac{60\!\cdots\!55}{62\!\cdots\!87}a^{13}+\frac{62\!\cdots\!08}{62\!\cdots\!87}a^{12}+\frac{22\!\cdots\!34}{62\!\cdots\!87}a^{11}+\frac{61\!\cdots\!00}{62\!\cdots\!87}a^{10}-\frac{37\!\cdots\!72}{62\!\cdots\!87}a^{9}-\frac{30\!\cdots\!70}{62\!\cdots\!87}a^{8}-\frac{12\!\cdots\!80}{62\!\cdots\!87}a^{7}+\frac{10\!\cdots\!75}{62\!\cdots\!87}a^{6}+\frac{89\!\cdots\!30}{60\!\cdots\!29}a^{5}+\frac{68\!\cdots\!91}{62\!\cdots\!87}a^{4}-\frac{22\!\cdots\!98}{62\!\cdots\!87}a^{3}+\frac{13\!\cdots\!35}{62\!\cdots\!87}a^{2}+\frac{17\!\cdots\!87}{60\!\cdots\!29}a+\frac{14\!\cdots\!20}{62\!\cdots\!87}$ | 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: | \( 432952.23952 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 432952.23952 \cdot 1}{2\cdot\sqrt{34895459505131153638432321}}\cr\approx \mathstrut & 0.17803052664 \end{aligned}\]
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 | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $17$ | ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $17$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1559\) | $\Q_{1559}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |