Normalized defining polynomial
\( x^{18} - 9 x^{17} + 36 x^{16} - 84 x^{15} + 117 x^{14} - 63 x^{13} - 126 x^{12} + 405 x^{11} - 549 x^{10} + 248 x^{9} + 441 x^{8} - 981 x^{7} + 834 x^{6} - 171 x^{5} - 783 x^{4} + \cdots + 166 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(808432815650446861638683444433\) \(\medspace = 3^{24}\cdot 17^{15}\) | 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: | \(45.87\) | 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: | $3^{25/18}17^{7/8}\approx 54.86700011448388$ | ||
Ramified primes: | \(3\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{4}$, $\frac{1}{12}a^{14}+\frac{1}{12}a^{13}-\frac{1}{12}a^{11}-\frac{1}{12}a^{10}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{12}a^{5}-\frac{5}{12}a^{4}+\frac{5}{12}a^{2}+\frac{5}{12}a-\frac{1}{2}$, $\frac{1}{12}a^{15}-\frac{1}{12}a^{13}-\frac{1}{12}a^{12}+\frac{1}{12}a^{10}-\frac{1}{12}a^{9}-\frac{1}{4}a^{7}+\frac{1}{12}a^{6}-\frac{1}{2}a^{5}+\frac{5}{12}a^{4}+\frac{5}{12}a^{3}+\frac{1}{12}a+\frac{1}{6}$, $\frac{1}{34116}a^{16}-\frac{2}{8529}a^{15}+\frac{61}{8529}a^{14}-\frac{392}{8529}a^{13}-\frac{691}{8529}a^{12}+\frac{622}{8529}a^{11}-\frac{553}{5686}a^{10}-\frac{236}{2843}a^{9}+\frac{408}{2843}a^{8}-\frac{3806}{8529}a^{7}-\frac{3157}{17058}a^{6}-\frac{6007}{17058}a^{5}-\frac{2591}{8529}a^{4}+\frac{368}{8529}a^{3}-\frac{247}{17058}a^{2}-\frac{3687}{11372}a+\frac{1689}{5686}$, $\frac{1}{3445716}a^{17}+\frac{7}{574286}a^{16}-\frac{2869}{574286}a^{15}+\frac{46189}{1148572}a^{14}+\frac{325385}{3445716}a^{13}+\frac{80677}{574286}a^{12}+\frac{470771}{3445716}a^{11}-\frac{404701}{3445716}a^{10}-\frac{258833}{1722858}a^{9}-\frac{1058303}{3445716}a^{8}+\frac{36991}{1148572}a^{7}-\frac{51475}{287143}a^{6}-\frac{411227}{1148572}a^{5}+\frac{930359}{3445716}a^{4}-\frac{115683}{287143}a^{3}+\frac{148435}{1722858}a^{2}-\frac{1410031}{3445716}a+\frac{79927}{1722858}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $9$ | 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: | $\frac{98949}{1148572}a^{17}-\frac{2489213}{3445716}a^{16}+\frac{3058937}{1148572}a^{15}-\frac{19430723}{3445716}a^{14}+\frac{11603843}{1722858}a^{13}-\frac{1731997}{1148572}a^{12}-\frac{39898051}{3445716}a^{11}+\frac{7993649}{287143}a^{10}-\frac{106098707}{3445716}a^{9}+\frac{4152885}{1148572}a^{8}+\frac{33769096}{861429}a^{7}-\frac{69558691}{1148572}a^{6}+\frac{125602357}{3445716}a^{5}+\frac{8972729}{1722858}a^{4}-\frac{71695157}{1148572}a^{3}+\frac{61042139}{861429}a^{2}+\frac{42548249}{1148572}a-\frac{40262251}{1722858}$, $\frac{48455}{3445716}a^{17}-\frac{106262}{861429}a^{16}+\frac{409325}{861429}a^{15}-\frac{1798091}{1722858}a^{14}+\frac{739693}{574286}a^{13}-\frac{289247}{861429}a^{12}-\frac{609741}{287143}a^{11}+\frac{8887925}{1722858}a^{10}-\frac{5003107}{861429}a^{9}+\frac{1473751}{1722858}a^{8}+\frac{6343417}{861429}a^{7}-\frac{19745801}{1722858}a^{6}+\frac{11692345}{1722858}a^{5}+\frac{746291}{574286}a^{4}-\frac{19224397}{1722858}a^{3}+\frac{14918721}{1148572}a^{2}+\frac{4851499}{861429}a-\frac{4883194}{861429}$, $\frac{1989}{287143}a^{17}-\frac{30671}{861429}a^{16}+\frac{11464}{287143}a^{15}+\frac{138229}{861429}a^{14}-\frac{659725}{861429}a^{13}+\frac{1350728}{861429}a^{12}-\frac{511572}{287143}a^{11}+\frac{397786}{861429}a^{10}+\frac{2820097}{861429}a^{9}-\frac{2131941}{287143}a^{8}+\frac{6747703}{861429}a^{7}-\frac{668106}{287143}a^{6}-\frac{4798844}{861429}a^{5}+\frac{7744256}{861429}a^{4}-\frac{9905587}{861429}a^{3}-\frac{473338}{287143}a^{2}+\frac{6912478}{861429}a-\frac{1262429}{861429}$, $\frac{22625}{3445716}a^{17}-\frac{18631}{287143}a^{16}+\frac{962587}{3445716}a^{15}-\frac{594260}{861429}a^{14}+\frac{3475373}{3445716}a^{13}-\frac{2013163}{3445716}a^{12}-\frac{578661}{574286}a^{11}+\frac{11475323}{3445716}a^{10}-\frac{15700895}{3445716}a^{9}+\frac{1656722}{861429}a^{8}+\frac{4467659}{1148572}a^{7}-\frac{28862855}{3445716}a^{6}+\frac{10828025}{1722858}a^{5}-\frac{690085}{3445716}a^{4}-\frac{19311151}{3445716}a^{3}+\frac{14344165}{1148572}a^{2}+\frac{11883185}{3445716}a-\frac{7994215}{1722858}$, $\frac{100595}{1148572}a^{17}-\frac{2118803}{3445716}a^{16}+\frac{535715}{287143}a^{15}-\frac{1780779}{574286}a^{14}+\frac{1162983}{574286}a^{13}+\frac{913376}{287143}a^{12}-\frac{9736559}{861429}a^{11}+\frac{4995206}{287143}a^{10}-\frac{7649381}{861429}a^{9}-\frac{8697855}{574286}a^{8}+\frac{31442077}{861429}a^{7}-\frac{8969511}{287143}a^{6}+\frac{7920}{287143}a^{5}+\frac{12832329}{574286}a^{4}-\frac{37290285}{574286}a^{3}+\frac{25315759}{3445716}a^{2}+\frac{40274629}{1148572}a-\frac{20422813}{1722858}$, $\frac{326483}{3445716}a^{17}-\frac{2814647}{3445716}a^{16}+\frac{10638409}{3445716}a^{15}-\frac{23090875}{3445716}a^{14}+\frac{7168685}{861429}a^{13}-\frac{8640313}{3445716}a^{12}-\frac{44656943}{3445716}a^{11}+\frac{56609239}{1722858}a^{10}-\frac{131483689}{3445716}a^{9}+\frac{25238423}{3445716}a^{8}+\frac{76567841}{1722858}a^{7}-\frac{252535847}{3445716}a^{6}+\frac{160769393}{3445716}a^{5}+\frac{3649742}{861429}a^{4}-\frac{243429721}{3445716}a^{3}+\frac{78069532}{861429}a^{2}+\frac{136903091}{3445716}a-\frac{57475547}{1722858}$, $\frac{18659}{3445716}a^{17}-\frac{146431}{3445716}a^{16}+\frac{666431}{3445716}a^{15}-\frac{1013773}{1722858}a^{14}+\frac{1441961}{1148572}a^{13}-\frac{6544499}{3445716}a^{12}+\frac{1532206}{861429}a^{11}-\frac{238311}{1148572}a^{10}-\frac{10444859}{3445716}a^{9}+\frac{10879237}{1722858}a^{8}-\frac{22817749}{3445716}a^{7}+\frac{9216743}{3445716}a^{6}+\frac{6610109}{1722858}a^{5}-\frac{8510777}{1148572}a^{4}+\frac{29799685}{3445716}a^{3}-\frac{1656095}{3445716}a^{2}-\frac{3225565}{574286}a+\frac{1842883}{861429}$, $\frac{307171}{3445716}a^{17}-\frac{1329641}{1722858}a^{16}+\frac{2646173}{861429}a^{15}-\frac{6430354}{861429}a^{14}+\frac{10213600}{861429}a^{13}-\frac{9627502}{861429}a^{12}-\frac{59137}{1722858}a^{11}+\frac{6436240}{287143}a^{10}-\frac{36387931}{861429}a^{9}+\frac{32073928}{861429}a^{8}-\frac{2779607}{1722858}a^{7}-\frac{71551493}{1722858}a^{6}+\frac{53559848}{861429}a^{5}-\frac{41374088}{861429}a^{4}-\frac{19556501}{1722858}a^{3}+\frac{156099157}{3445716}a^{2}-\frac{4160014}{287143}a-\frac{164494}{861429}$, $\frac{307171}{3445716}a^{17}-\frac{2562625}{3445716}a^{16}+\frac{2452859}{861429}a^{15}-\frac{5702851}{861429}a^{14}+\frac{8504074}{861429}a^{13}-\frac{2405342}{287143}a^{12}-\frac{2140747}{1722858}a^{11}+\frac{32172125}{1722858}a^{10}-\frac{28441049}{861429}a^{9}+\frac{23752033}{861429}a^{8}+\frac{2296249}{1722858}a^{7}-\frac{28123027}{861429}a^{6}+\frac{74369941}{1722858}a^{5}-\frac{31660817}{861429}a^{4}-\frac{3862803}{574286}a^{3}+\frac{115242031}{3445716}a^{2}-\frac{25602701}{3445716}a-\frac{5602501}{1722858}$ (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)]
| |
Regulator: | \( 250225810.428 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 250225810.428 \cdot 1}{2\cdot\sqrt{808432815650446861638683444433}}\cr\approx \mathstrut & 1.35200743553 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 72 |
The 9 conjugacy class representatives for $F_9$ |
Character table for $F_9$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 9.1.218070794717793.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 24 sibling: | data not computed |
Degree 36 sibling: | data not computed |
Minimal sibling: | 9.1.218070794717793.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $18$ | $9$ | $2$ | $24$ | |||
\(17\) | 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.8.7.1 | $x^{8} + 68$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
17.8.7.1 | $x^{8} + 68$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |