Normalized defining polynomial
\( x^{17} - 5 x^{15} - x^{14} + 5 x^{13} + 6 x^{12} + 9 x^{11} - 16 x^{10} - 21 x^{9} + 19 x^{8} + 13 x^{7} + \cdots + 1 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(3621113810274499901\) \(\medspace = 55954279\cdot 64715583419\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.35\) | 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: | $55954279^{1/2}64715583419^{1/2}\approx 1902922439.3743691$ | ||
Ramified primes: | \(55954279\), \(64715583419\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{36211\!\cdots\!99901}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1018643}a^{16}+\frac{36148}{1018643}a^{15}-\frac{241070}{1018643}a^{14}+\frac{292504}{1018643}a^{13}-\frac{79743}{1018643}a^{12}+\frac{209732}{1018643}a^{11}-\frac{367504}{1018643}a^{10}-\frac{411245}{1018643}a^{9}+\frac{391661}{1018643}a^{8}-\frac{357210}{1018643}a^{7}-\frac{108399}{1018643}a^{6}+\frac{312564}{1018643}a^{5}-\frac{224688}{1018643}a^{4}-\frac{381190}{1018643}a^{3}-\frac{72256}{1018643}a^{2}-\frac{109236}{1018643}a-\frac{402661}{1018643}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $10$ | 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: | $a$, $\frac{170896}{1018643}a^{16}-\frac{521187}{1018643}a^{15}-\frac{919871}{1018643}a^{14}+\frac{1932931}{1018643}a^{13}+\frac{1683612}{1018643}a^{12}+\frac{387274}{1018643}a^{11}-\frac{1548062}{1018643}a^{10}-\frac{8019522}{1018643}a^{9}+\frac{2341298}{1018643}a^{8}+\frac{10682617}{1018643}a^{7}-\frac{932549}{1018643}a^{6}-\frac{4757505}{1018643}a^{5}-\frac{3588492}{1018643}a^{4}+\frac{1429539}{1018643}a^{3}+\frac{2784999}{1018643}a^{2}-\frac{343838}{1018643}a+\frac{254966}{1018643}$, $\frac{294536}{1018643}a^{16}+\frac{30692}{1018643}a^{15}-\frac{1320491}{1018643}a^{14}-\frac{810867}{1018643}a^{13}+\frac{686046}{1018643}a^{12}+\frac{3112832}{1018643}a^{11}+\frac{3958894}{1018643}a^{10}-\frac{3692762}{1018643}a^{9}-\frac{7130026}{1018643}a^{8}-\frac{662305}{1018643}a^{7}+\frac{5012900}{1018643}a^{6}+\frac{5563751}{1018643}a^{5}-\frac{1543630}{1018643}a^{4}-\frac{3420952}{1018643}a^{3}-\frac{503660}{1018643}a^{2}-\frac{95341}{1018643}a+\frac{406908}{1018643}$, $\frac{528514}{1018643}a^{16}+\frac{74607}{1018643}a^{15}-\frac{2096755}{1018643}a^{14}-\frac{877196}{1018643}a^{13}+\frac{43580}{1018643}a^{12}+\frac{2660203}{1018643}a^{11}+\frac{6894113}{1018643}a^{10}-\frac{3938949}{1018643}a^{9}-\frac{6234134}{1018643}a^{8}+\frac{1751751}{1018643}a^{7}-\frac{1906766}{1018643}a^{6}+\frac{3151872}{1018643}a^{5}+\frac{2647308}{1018643}a^{4}-\frac{2132335}{1018643}a^{3}-\frac{400157}{1018643}a^{2}-\frac{1163279}{1018643}a-\frac{136123}{1018643}$, $\frac{470725}{1018643}a^{16}+\frac{354628}{1018643}a^{15}-\frac{1864193}{1018643}a^{14}-\frac{2047553}{1018643}a^{13}-\frac{29125}{1018643}a^{12}+\frac{3290712}{1018643}a^{11}+\frac{6894862}{1018643}a^{10}-\frac{1405548}{1018643}a^{9}-\frac{9758775}{1018643}a^{8}-\frac{2314526}{1018643}a^{7}+\frac{4839096}{1018643}a^{6}+\frac{3987195}{1018643}a^{5}+\frac{462533}{1018643}a^{4}-\frac{2716943}{1018643}a^{3}-\frac{2253116}{1018643}a^{2}-\frac{36103}{1018643}a-\frac{640286}{1018643}$, $\frac{18707}{1018643}a^{16}-\frac{158316}{1018643}a^{15}-\frac{163929}{1018643}a^{14}+\frac{740775}{1018643}a^{13}+\frac{559694}{1018643}a^{12}-\frac{356312}{1018643}a^{11}-\frac{1094364}{1018643}a^{10}-\frac{2405565}{1018643}a^{9}+\frac{1740514}{1018643}a^{8}+\frac{4045182}{1018643}a^{7}-\frac{720523}{1018643}a^{6}-\frac{2932001}{1018643}a^{5}-\frac{2354684}{1018643}a^{4}+\frac{1616956}{1018643}a^{3}+\frac{2083555}{1018643}a^{2}-\frac{79994}{1018643}a+\frac{285658}{1018643}$, $\frac{48090}{1018643}a^{16}-\frac{466281}{1018643}a^{15}+\frac{119683}{1018643}a^{14}+\frac{2113459}{1018643}a^{13}-\frac{668618}{1018643}a^{12}-\frac{1609749}{1018643}a^{11}-\frac{1848596}{1018643}a^{10}-\frac{3892777}{1018643}a^{9}+\frac{8417564}{1018643}a^{8}+\frac{5259867}{1018643}a^{7}-\frac{9679466}{1018643}a^{6}+\frac{106652}{1018643}a^{5}+\frac{1537667}{1018643}a^{4}+\frac{72328}{1018643}a^{3}+\frac{1837519}{1018643}a^{2}-\frac{1035932}{1018643}a+\frac{435940}{1018643}$, $\frac{313243}{1018643}a^{16}+\frac{891019}{1018643}a^{15}-\frac{1484420}{1018643}a^{14}-\frac{4144664}{1018643}a^{13}+\frac{227097}{1018643}a^{12}+\frac{3775163}{1018643}a^{11}+\frac{7957745}{1018643}a^{10}+\frac{4088103}{1018643}a^{9}-\frac{16594585}{1018643}a^{8}-\frac{7822196}{1018643}a^{7}+\frac{12441521}{1018643}a^{6}+\frac{4669036}{1018643}a^{5}-\frac{842385}{1018643}a^{4}-\frac{3841282}{1018643}a^{3}-\frac{1476034}{1018643}a^{2}+\frac{843308}{1018643}a-\frac{326077}{1018643}$, $\frac{299438}{1018643}a^{16}-\frac{15694}{1018643}a^{15}-\frac{1419751}{1018643}a^{14}-\frac{186960}{1018643}a^{13}+\frac{944772}{1018643}a^{12}+\frac{1371023}{1018643}a^{11}+\frac{3415110}{1018643}a^{10}-\frac{3721255}{1018643}a^{9}-\frac{4293930}{1018643}a^{8}+\frac{4434807}{1018643}a^{7}-\frac{739210}{1018643}a^{6}-\frac{1417094}{1018643}a^{5}+\frac{1244806}{1018643}a^{4}+\frac{251502}{1018643}a^{3}+\frac{803835}{1018643}a^{2}-\frac{1801281}{1018643}a-\frac{325823}{1018643}$, $\frac{236287}{1018643}a^{16}-\frac{19079}{1018643}a^{15}-\frac{1227816}{1018643}a^{14}-\frac{34902}{1018643}a^{13}+\frac{1642616}{1018643}a^{12}+\frac{981777}{1018643}a^{11}+\frac{860816}{1018643}a^{10}-\frac{4510188}{1018643}a^{9}-\frac{4407058}{1018643}a^{8}+\frac{7810211}{1018643}a^{7}+\frac{4578294}{1018643}a^{6}-\frac{4956776}{1018643}a^{5}-\frac{3254868}{1018643}a^{4}+\frac{209816}{1018643}a^{3}+\frac{2359137}{1018643}a^{2}-\frac{670398}{1018643}a-\frac{266221}{1018643}$ | 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: | \( 284.067800863 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{6}\cdot 284.067800863 \cdot 1}{2\cdot\sqrt{3621113810274499901}}\cr\approx \mathstrut & 0.146960328672 \end{aligned}\]
Galois group
A non-solvable group of order 355687428096000 |
The 297 conjugacy class representatives for $S_{17}$ |
Character table for $S_{17}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{3}$ | $17$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.7.0.1}{7} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.8.0.1}{8} }$ | $15{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(55954279\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(64715583419\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |