Normalized defining polynomial
\( x^{18} - 36 x^{16} + 486 x^{14} - 3345 x^{12} + 13113 x^{10} - 30537 x^{8} + 42276 x^{6} - 33462 x^{4} + 13689 x^{2} - 2197 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(45459928664503948293335446409613\) \(\medspace = 3^{36}\cdot 13^{13}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(57.38\) | 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: | $3^{37/18}13^{3/4}\approx 65.49479091539757$ | ||
Ramified primes: | \(3\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{26}a^{12}+\frac{3}{26}a^{10}+\frac{5}{26}a^{8}-\frac{2}{13}a^{6}-\frac{2}{13}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{26}a^{13}+\frac{3}{26}a^{11}+\frac{5}{26}a^{9}-\frac{2}{13}a^{7}-\frac{2}{13}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{26}a^{14}-\frac{2}{13}a^{10}+\frac{7}{26}a^{8}+\frac{4}{13}a^{6}-\frac{1}{2}a^{5}-\frac{1}{26}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{338}a^{15}+\frac{3}{338}a^{13}+\frac{57}{338}a^{11}-\frac{28}{169}a^{9}-\frac{41}{169}a^{7}-\frac{1}{2}a^{6}-\frac{9}{26}a^{5}+\frac{1}{26}a^{3}$, $\frac{1}{1626794}a^{16}+\frac{6781}{813397}a^{14}+\frac{25719}{1626794}a^{12}-\frac{49831}{813397}a^{10}-\frac{446775}{1626794}a^{8}-\frac{1}{2}a^{7}-\frac{15577}{125138}a^{6}-\frac{1}{2}a^{5}-\frac{15584}{62569}a^{4}-\frac{1}{2}a^{3}+\frac{668}{4813}a^{2}+\frac{2747}{9626}$, $\frac{1}{1626794}a^{17}-\frac{877}{1626794}a^{15}-\frac{8799}{813397}a^{13}-\frac{54644}{813397}a^{11}+\frac{361809}{1626794}a^{9}+\frac{84050}{813397}a^{7}-\frac{1}{2}a^{6}-\frac{26355}{125138}a^{5}-\frac{1}{2}a^{4}-\frac{29820}{62569}a^{3}-\frac{1}{2}a^{2}+\frac{2747}{9626}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $17$ | 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: | $\frac{11689}{813397}a^{17}-\frac{1217}{1626794}a^{16}-\frac{31436}{62569}a^{15}+\frac{75831}{1626794}a^{14}+\frac{5274833}{813397}a^{13}-\frac{1642317}{1626794}a^{12}-\frac{68437991}{1626794}a^{11}+\frac{16235303}{1626794}a^{10}+\frac{124581993}{813397}a^{9}-\frac{82590515}{1626794}a^{8}-\frac{529292001}{1626794}a^{7}+\frac{17282285}{125138}a^{6}+\frac{49659913}{125138}a^{5}-\frac{24777121}{125138}a^{4}-\frac{31535165}{125138}a^{3}+\frac{664635}{4813}a^{2}+\frac{596319}{9626}a-\frac{172300}{4813}$, $\frac{28142}{813397}a^{17}+\frac{41775}{813397}a^{16}-\frac{148381}{125138}a^{15}-\frac{2835095}{1626794}a^{14}+\frac{24022373}{1626794}a^{13}+\frac{17370539}{813397}a^{12}-\frac{73416390}{813397}a^{11}-\frac{207493815}{1626794}a^{10}+\frac{485429221}{1626794}a^{9}+\frac{331985344}{813397}a^{8}-\frac{443733157}{813397}a^{7}-\frac{44560703}{62569}a^{6}+\frac{33518622}{62569}a^{5}+\frac{40807865}{62569}a^{4}-\frac{32356839}{125138}a^{3}-\frac{1357414}{4813}a^{2}+\frac{230692}{4813}a+\frac{211338}{4813}$, $\frac{5488}{813397}a^{17}-\frac{6001}{813397}a^{16}-\frac{418683}{1626794}a^{15}+\frac{204414}{813397}a^{14}+\frac{3046463}{813397}a^{13}-\frac{5032081}{1626794}a^{12}-\frac{45161477}{1626794}a^{11}+\frac{15053720}{813397}a^{10}+\frac{14351229}{125138}a^{9}-\frac{47912729}{813397}a^{8}-\frac{217118218}{813397}a^{7}+\frac{12681237}{125138}a^{6}+\frac{20878017}{62569}a^{5}-\frac{11649923}{125138}a^{4}-\frac{12452299}{62569}a^{3}+\frac{440227}{9626}a^{2}+\frac{217805}{4813}a-\frac{82265}{9626}$, $\frac{158515}{1626794}a^{17}+\frac{138207}{1626794}a^{16}-\frac{5620547}{1626794}a^{15}-\frac{4896581}{1626794}a^{14}+\frac{73923315}{1626794}a^{13}+\frac{64321875}{1626794}a^{12}-\frac{243985538}{813397}a^{11}-\frac{211944390}{813397}a^{10}+\frac{894470998}{813397}a^{9}+\frac{1550871839}{1626794}a^{8}-\frac{1866815518}{813397}a^{7}-\frac{248461851}{125138}a^{6}+\frac{330212227}{125138}a^{5}+\frac{285309051}{125138}a^{4}-\frac{93370005}{62569}a^{3}-\frac{6214273}{4813}a^{2}+\frac{1455417}{4813}a+\frac{1266007}{4813}$, $\frac{36917}{1626794}a^{17}-\frac{1255351}{1626794}a^{15}+\frac{7726365}{813397}a^{13}-\frac{46590446}{813397}a^{11}+\frac{900444}{4813}a^{9}-\frac{277214229}{813397}a^{7}+\frac{21237570}{62569}a^{5}-\frac{10556798}{62569}a^{3}+\frac{1}{2}a^{2}+\frac{152154}{4813}a-\frac{1}{2}$, $\frac{4611}{1626794}a^{16}-\frac{79878}{813397}a^{14}+\frac{1012131}{813397}a^{12}-\frac{6336842}{813397}a^{10}+\frac{21651274}{813397}a^{8}-\frac{3201219}{62569}a^{6}+\frac{3484878}{62569}a^{4}-\frac{168627}{4813}a^{2}+\frac{1}{2}a+\frac{49837}{4813}$, $\frac{6593}{813397}a^{17}-\frac{83071}{1626794}a^{16}-\frac{227859}{813397}a^{15}+\frac{1443343}{813397}a^{14}+\frac{2876738}{813397}a^{13}-\frac{36812547}{1626794}a^{12}-\frac{2757973}{125138}a^{11}+\frac{233461999}{1626794}a^{10}+\frac{60775867}{813397}a^{9}-\frac{408047586}{813397}a^{8}-\frac{228827253}{1626794}a^{7}+\frac{124543853}{125138}a^{6}+\frac{8899648}{62569}a^{5}-\frac{68429445}{62569}a^{4}-\frac{4417193}{62569}a^{3}+\frac{2895075}{4813}a^{2}+\frac{138881}{9626}a-\frac{1176453}{9626}$, $\frac{8575}{1626794}a^{17}+\frac{383}{62569}a^{16}-\frac{147981}{813397}a^{15}-\frac{13420}{62569}a^{14}+\frac{1864607}{813397}a^{13}+\frac{347681}{125138}a^{12}-\frac{23179965}{1626794}a^{11}-\frac{2264209}{125138}a^{10}+\frac{39124593}{813397}a^{9}+\frac{316683}{4813}a^{8}-\frac{145595559}{1626794}a^{7}-\frac{17179999}{125138}a^{6}+\frac{5414770}{62569}a^{5}+\frac{9924493}{62569}a^{4}-\frac{2225042}{62569}a^{3}-\frac{860771}{9626}a^{2}+\frac{12384}{4813}a+\frac{165989}{9626}$, $\frac{5488}{813397}a^{17}+\frac{6001}{813397}a^{16}-\frac{418683}{1626794}a^{15}-\frac{204414}{813397}a^{14}+\frac{3046463}{813397}a^{13}+\frac{5032081}{1626794}a^{12}-\frac{45161477}{1626794}a^{11}-\frac{15053720}{813397}a^{10}+\frac{14351229}{125138}a^{9}+\frac{47912729}{813397}a^{8}-\frac{217118218}{813397}a^{7}-\frac{12681237}{125138}a^{6}+\frac{20878017}{62569}a^{5}+\frac{11649923}{125138}a^{4}-\frac{12452299}{62569}a^{3}-\frac{440227}{9626}a^{2}+\frac{217805}{4813}a+\frac{82265}{9626}$, $\frac{4413}{125138}a^{17}+\frac{97393}{1626794}a^{16}-\frac{154565}{125138}a^{15}-\frac{3303881}{1626794}a^{14}+\frac{997588}{62569}a^{13}+\frac{40507933}{1626794}a^{12}-\frac{12851989}{125138}a^{11}-\frac{121409958}{813397}a^{10}+\frac{1757983}{4813}a^{9}+\frac{785719011}{1626794}a^{8}-\frac{45960532}{62569}a^{7}-\frac{54131546}{62569}a^{6}+\frac{50450185}{62569}a^{5}+\frac{104210713}{125138}a^{4}-\frac{4160473}{9626}a^{3}-\frac{1871054}{4813}a^{2}+\frac{414210}{4813}a+\frac{648095}{9626}$, $\frac{11127}{1626794}a^{17}+\frac{13151}{813397}a^{16}-\frac{193734}{813397}a^{15}-\frac{468340}{813397}a^{14}+\frac{2463829}{813397}a^{13}+\frac{12416341}{1626794}a^{12}-\frac{15312464}{813397}a^{11}-\frac{41505366}{813397}a^{10}+\frac{50195004}{813397}a^{9}+\frac{310162373}{1626794}a^{8}-\frac{12830981}{125138}a^{7}-\frac{51123315}{125138}a^{6}+\frac{8089845}{125138}a^{5}+\frac{30446189}{62569}a^{4}+\frac{181209}{9626}a^{3}-\frac{2801407}{9626}a^{2}-\frac{246957}{9626}a+\frac{658219}{9626}$, $\frac{13388}{813397}a^{17}+\frac{198}{62569}a^{16}-\frac{493295}{813397}a^{15}-\frac{5191}{62569}a^{14}+\frac{13681637}{1626794}a^{13}+\frac{33899}{62569}a^{12}-\frac{95755749}{1626794}a^{11}+\frac{72643}{125138}a^{10}+\frac{185569460}{813397}a^{9}-\frac{1066996}{62569}a^{8}-\frac{808381659}{1626794}a^{7}+\frac{3727167}{62569}a^{6}+\frac{36511708}{62569}a^{5}-\frac{5056998}{62569}a^{4}-\frac{41171333}{125138}a^{3}+\frac{404243}{9626}a^{2}+\frac{641535}{9626}a-\frac{71233}{9626}$, $\frac{1471}{1626794}a^{17}+\frac{24316}{813397}a^{16}-\frac{45815}{813397}a^{15}-\frac{1619439}{1626794}a^{14}+\frac{152725}{125138}a^{13}+\frac{19220781}{1626794}a^{12}-\frac{19621423}{1626794}a^{11}-\frac{54706559}{813397}a^{10}+\frac{49633812}{813397}a^{9}+\frac{163317091}{813397}a^{8}-\frac{264515255}{1626794}a^{7}-\frac{19951556}{62569}a^{6}+\frac{27506805}{125138}a^{5}+\frac{2520587}{9626}a^{4}-\frac{16704999}{125138}a^{3}-\frac{1018691}{9626}a^{2}+\frac{267445}{9626}a+\frac{161305}{9626}$, $\frac{4865}{1626794}a^{17}+\frac{22305}{1626794}a^{16}-\frac{175555}{1626794}a^{15}-\frac{771533}{1626794}a^{14}+\frac{1183685}{813397}a^{13}+\frac{9790467}{1626794}a^{12}-\frac{8029902}{813397}a^{11}-\frac{62015357}{1626794}a^{10}+\frac{59979657}{1626794}a^{9}+\frac{109175748}{813397}a^{8}-\frac{61538622}{813397}a^{7}-\frac{33964669}{125138}a^{6}+\frac{4849720}{62569}a^{5}+\frac{19202423}{62569}a^{4}-\frac{1873111}{62569}a^{3}-\frac{848396}{4813}a^{2}+\frac{4040}{4813}a+\frac{191286}{4813}$, $\frac{39401}{813397}a^{17}-\frac{21981}{1626794}a^{16}-\frac{1344977}{813397}a^{15}+\frac{785791}{1626794}a^{14}+\frac{33284363}{1626794}a^{13}-\frac{5202439}{813397}a^{12}-\frac{100803624}{813397}a^{11}+\frac{68392511}{1626794}a^{10}+\frac{658485135}{1626794}a^{9}-\frac{122050179}{813397}a^{8}-\frac{592842052}{813397}a^{7}+\frac{18398301}{62569}a^{6}+\frac{43991841}{62569}a^{5}-\frac{19015095}{62569}a^{4}-\frac{41455539}{125138}a^{3}+\frac{718292}{4813}a^{2}+\frac{553101}{9626}a-\frac{248185}{9626}$, $\frac{10737}{813397}a^{17}+\frac{53375}{1626794}a^{16}-\frac{27190}{62569}a^{15}-\frac{1866081}{1626794}a^{14}+\frac{4109330}{813397}a^{13}+\frac{24014261}{1626794}a^{12}-\frac{22579200}{813397}a^{11}-\frac{77006443}{813397}a^{10}+\frac{63129786}{813397}a^{9}+\frac{272610383}{813397}a^{8}-\frac{173878729}{1626794}a^{7}-\frac{6484645}{9626}a^{6}+\frac{3635702}{62569}a^{5}+\frac{93773877}{125138}a^{4}+\frac{154547}{62569}a^{3}-\frac{1997599}{4813}a^{2}-\frac{33216}{4813}a+\frac{400832}{4813}$, $\frac{33589}{1626794}a^{17}+\frac{17963}{813397}a^{16}-\frac{1108983}{1626794}a^{15}-\frac{592601}{813397}a^{14}+\frac{12956343}{1626794}a^{13}+\frac{6926060}{813397}a^{12}-\frac{71756549}{1626794}a^{11}-\frac{38546782}{813397}a^{10}+\frac{203731043}{1626794}a^{9}+\frac{111053435}{813397}a^{8}-\frac{145815795}{813397}a^{7}-\frac{12702590}{62569}a^{6}+\frac{6970502}{62569}a^{5}+\frac{17960185}{125138}a^{4}-\frac{1285423}{125138}a^{3}-\frac{349449}{9626}a^{2}-\frac{92487}{9626}a-\frac{11469}{9626}$ (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: | \( 40148447797.4 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 40148447797.4 \cdot 1}{2\cdot\sqrt{45459928664503948293335446409613}}\cr\approx \mathstrut & 0.780484550563 \end{aligned}\] (assuming GRH)
Galois group
$C_3^2:C_4$ (as 18T10):
A solvable group of order 36 |
The 6 conjugacy class representatives for $C_3^2 : C_4$ |
Character table for $C_3^2 : C_4$ |
Intermediate fields
\(\Q(\sqrt{13}) \), 6.6.187388721.1 x2, 6.6.1686498489.1 x2, 9.9.1870004703089601.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 6 siblings: | 6.6.1686498489.1, 6.6.187388721.1 |
Degree 9 sibling: | 9.9.1870004703089601.1 |
Degree 12 siblings: | deg 12, deg 12 |
Minimal sibling: | 6.6.187388721.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.18.2 | $x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$ | $9$ | $1$ | $18$ | $C_3^2:C_2$ | $[3/2, 5/2]_{2}$ |
3.9.18.2 | $x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$ | $9$ | $1$ | $18$ | $C_3^2:C_2$ | $[3/2, 5/2]_{2}$ | |
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |