Normalized defining polynomial
\( x^{18} - 3 x^{17} - 9 x^{15} + 30 x^{14} + 12 x^{13} - 87 x^{11} - 39 x^{10} + 133 x^{9} + 177 x^{8} + \cdots + 20 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(2261987535219532470703125\) \(\medspace = 3^{32}\cdot 5^{13}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.54\) | 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: | $3^{16/9}5^{3/4}\approx 23.574531309937974$ | ||
Ramified primes: | \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{10}a^{14}+\frac{1}{10}a^{12}+\frac{1}{10}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{10}a^{8}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{10}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{10}a^{15}+\frac{1}{10}a^{13}+\frac{1}{10}a^{12}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{1}{10}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{10}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{100}a^{16}+\frac{1}{100}a^{15}-\frac{1}{20}a^{14}-\frac{9}{50}a^{13}-\frac{17}{100}a^{12}-\frac{6}{25}a^{11}-\frac{43}{100}a^{10}-\frac{3}{100}a^{9}-\frac{11}{25}a^{8}-\frac{13}{50}a^{7}-\frac{1}{100}a^{6}-\frac{7}{25}a^{5}-\frac{7}{50}a^{4}-\frac{1}{5}a^{3}+\frac{9}{20}a^{2}-\frac{1}{5}$, $\frac{1}{81\!\cdots\!00}a^{17}-\frac{1107977017497}{20\!\cdots\!50}a^{16}-\frac{28983675025203}{10\!\cdots\!75}a^{15}-\frac{141766698683313}{81\!\cdots\!00}a^{14}-\frac{53306222641}{56311595230360}a^{13}+\frac{16\!\cdots\!69}{81\!\cdots\!00}a^{12}+\frac{33\!\cdots\!63}{81\!\cdots\!00}a^{11}-\frac{758966312313633}{40\!\cdots\!00}a^{10}+\frac{13\!\cdots\!83}{81\!\cdots\!00}a^{9}-\frac{22305245965515}{163303626168044}a^{8}+\frac{38\!\cdots\!03}{81\!\cdots\!00}a^{7}-\frac{32\!\cdots\!79}{81\!\cdots\!00}a^{6}+\frac{361939030345039}{40\!\cdots\!00}a^{5}-\frac{255806080064867}{40\!\cdots\!00}a^{4}-\frac{143337695314541}{326607252336088}a^{3}+\frac{191678506413929}{16\!\cdots\!40}a^{2}+\frac{132720717096673}{816518130840220}a+\frac{119305148371769}{408259065420110}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{895324209}{90971882440}a^{17}-\frac{74373873}{2274297061}a^{16}-\frac{1046484}{2274297061}a^{15}-\frac{4074798717}{90971882440}a^{14}+\frac{24470964723}{90971882440}a^{13}+\frac{14315882601}{90971882440}a^{12}-\frac{42424997709}{90971882440}a^{11}-\frac{21703898457}{45485941220}a^{10}-\frac{6431230553}{18194376488}a^{9}+\frac{99887733261}{45485941220}a^{8}+\frac{81621349323}{90971882440}a^{7}-\frac{36470652939}{18194376488}a^{6}-\frac{101727166989}{45485941220}a^{5}-\frac{16681105827}{45485941220}a^{4}+\frac{49908097515}{18194376488}a^{3}+\frac{38695708137}{18194376488}a^{2}+\frac{2336125695}{9097188244}a-\frac{1176873135}{4548594122}$, $\frac{73050119606213}{81\!\cdots\!00}a^{17}-\frac{15528246347637}{816518130840220}a^{16}-\frac{137354966393399}{40\!\cdots\!00}a^{15}-\frac{392806565823639}{81\!\cdots\!00}a^{14}+\frac{54593310892987}{281557976151800}a^{13}+\frac{35\!\cdots\!79}{81\!\cdots\!00}a^{12}-\frac{17\!\cdots\!37}{81\!\cdots\!00}a^{11}-\frac{183498350663762}{204129532710055}a^{10}-\frac{75\!\cdots\!03}{81\!\cdots\!00}a^{9}+\frac{69\!\cdots\!97}{40\!\cdots\!00}a^{8}+\frac{51\!\cdots\!39}{16\!\cdots\!40}a^{7}-\frac{52\!\cdots\!21}{81\!\cdots\!00}a^{6}-\frac{17\!\cdots\!49}{40\!\cdots\!00}a^{5}-\frac{10\!\cdots\!89}{40\!\cdots\!00}a^{4}+\frac{37\!\cdots\!59}{16\!\cdots\!40}a^{3}+\frac{69\!\cdots\!63}{16\!\cdots\!40}a^{2}+\frac{800631627195709}{816518130840220}a-\frac{317395344215987}{408259065420110}$, $\frac{63805268336677}{40\!\cdots\!00}a^{17}-\frac{60483444442058}{10\!\cdots\!75}a^{16}+\frac{44935627984139}{10\!\cdots\!75}a^{15}-\frac{615146504238211}{40\!\cdots\!00}a^{14}+\frac{76408438543857}{140778988075900}a^{13}-\frac{185581691787323}{816518130840220}a^{12}-\frac{32959738564413}{816518130840220}a^{11}-\frac{944716302909526}{10\!\cdots\!75}a^{10}+\frac{16\!\cdots\!59}{40\!\cdots\!00}a^{9}+\frac{40\!\cdots\!37}{20\!\cdots\!50}a^{8}+\frac{682042760214307}{40\!\cdots\!00}a^{7}-\frac{74\!\cdots\!37}{40\!\cdots\!00}a^{6}-\frac{13\!\cdots\!29}{10\!\cdots\!75}a^{5}+\frac{851264780997914}{10\!\cdots\!75}a^{4}+\frac{503100160872669}{816518130840220}a^{3}+\frac{388847577664829}{816518130840220}a^{2}+\frac{17879568693668}{204129532710055}a-\frac{40427986026786}{204129532710055}$, $\frac{4927513986291}{81\!\cdots\!00}a^{17}-\frac{11982466796479}{40\!\cdots\!00}a^{16}+\frac{21033289075033}{40\!\cdots\!00}a^{15}-\frac{128801229122353}{81\!\cdots\!00}a^{14}+\frac{2857325212109}{56311595230360}a^{13}-\frac{623106434261991}{81\!\cdots\!00}a^{12}+\frac{929354105550113}{81\!\cdots\!00}a^{11}-\frac{494156491192729}{20\!\cdots\!50}a^{10}+\frac{29\!\cdots\!23}{81\!\cdots\!00}a^{9}-\frac{172599732015937}{816518130840220}a^{8}+\frac{802930770033173}{81\!\cdots\!00}a^{7}-\frac{63\!\cdots\!79}{81\!\cdots\!00}a^{6}-\frac{574058895365971}{40\!\cdots\!00}a^{5}+\frac{36\!\cdots\!13}{40\!\cdots\!00}a^{4}+\frac{443279370613781}{326607252336088}a^{3}-\frac{778685265916051}{16\!\cdots\!40}a^{2}-\frac{848387427251427}{816518130840220}a-\frac{397212777299321}{408259065420110}$, $\frac{1983840969917}{81\!\cdots\!00}a^{17}-\frac{77246277370691}{40\!\cdots\!00}a^{16}+\frac{342914221818883}{40\!\cdots\!00}a^{15}-\frac{10\!\cdots\!91}{81\!\cdots\!00}a^{14}+\frac{93134085244067}{281557976151800}a^{13}-\frac{15\!\cdots\!81}{16\!\cdots\!40}a^{12}+\frac{19\!\cdots\!83}{16\!\cdots\!40}a^{11}-\frac{27\!\cdots\!21}{20\!\cdots\!50}a^{10}+\frac{23\!\cdots\!29}{81\!\cdots\!00}a^{9}-\frac{11\!\cdots\!33}{40\!\cdots\!00}a^{8}+\frac{49\!\cdots\!07}{81\!\cdots\!00}a^{7}-\frac{15\!\cdots\!37}{81\!\cdots\!00}a^{6}+\frac{12\!\cdots\!87}{40\!\cdots\!00}a^{5}+\frac{25\!\cdots\!13}{40\!\cdots\!00}a^{4}-\frac{545637823551601}{16\!\cdots\!40}a^{3}-\frac{25\!\cdots\!61}{16\!\cdots\!40}a^{2}-\frac{280531890133729}{816518130840220}a-\frac{99099393197411}{408259065420110}$, $\frac{69900132713}{13190922953800}a^{17}-\frac{72862198808}{1648865369225}a^{16}+\frac{172155180276}{1648865369225}a^{15}-\frac{1283155236869}{13190922953800}a^{14}+\frac{33656224027}{90971882440}a^{13}-\frac{11867460997123}{13190922953800}a^{12}+\frac{1943382617319}{13190922953800}a^{11}+\frac{883267929081}{6595461476900}a^{10}+\frac{23718261993679}{13190922953800}a^{9}+\frac{150891444293}{1319092295380}a^{8}-\frac{48653636205161}{13190922953800}a^{7}-\frac{23137983818927}{13190922953800}a^{6}+\frac{26711014391117}{6595461476900}a^{5}+\frac{23067948957819}{6595461476900}a^{4}-\frac{767349106121}{527636918152}a^{3}-\frac{7590949822223}{2638184590760}a^{2}-\frac{1167144014751}{1319092295380}a-\frac{322001355053}{659546147690}$, $\frac{3761691433816}{10\!\cdots\!75}a^{17}-\frac{26394951724251}{10\!\cdots\!75}a^{16}+\frac{117313324706571}{20\!\cdots\!50}a^{15}-\frac{171501593980071}{20\!\cdots\!50}a^{14}+\frac{10076124301126}{35194747018975}a^{13}-\frac{109371680217024}{204129532710055}a^{12}+\frac{120928489198969}{408259065420110}a^{11}-\frac{559382958405247}{10\!\cdots\!75}a^{10}+\frac{12\!\cdots\!37}{10\!\cdots\!75}a^{9}+\frac{155481988115237}{10\!\cdots\!75}a^{8}-\frac{21\!\cdots\!43}{20\!\cdots\!50}a^{7}-\frac{26\!\cdots\!37}{20\!\cdots\!50}a^{6}+\frac{846460057812059}{20\!\cdots\!50}a^{5}+\frac{43\!\cdots\!51}{20\!\cdots\!50}a^{4}+\frac{128669901672662}{204129532710055}a^{3}-\frac{250917634345981}{408259065420110}a^{2}-\frac{410988934384639}{204129532710055}a-\frac{42921158480447}{204129532710055}$, $\frac{33393205855107}{10\!\cdots\!75}a^{17}-\frac{494517189470459}{40\!\cdots\!00}a^{16}+\frac{345191977323493}{40\!\cdots\!00}a^{15}-\frac{14\!\cdots\!39}{40\!\cdots\!00}a^{14}+\frac{17147027623609}{14077898807590}a^{13}-\frac{17\!\cdots\!93}{40\!\cdots\!00}a^{12}+\frac{204730740780806}{10\!\cdots\!75}a^{11}-\frac{11\!\cdots\!73}{40\!\cdots\!00}a^{10}+\frac{13\!\cdots\!19}{40\!\cdots\!00}a^{9}+\frac{19\!\cdots\!53}{408259065420110}a^{8}+\frac{22\!\cdots\!06}{10\!\cdots\!75}a^{7}-\frac{11\!\cdots\!97}{40\!\cdots\!00}a^{6}-\frac{15\!\cdots\!33}{20\!\cdots\!50}a^{5}-\frac{955080659509761}{20\!\cdots\!50}a^{4}+\frac{464170264962067}{81651813084022}a^{3}+\frac{34\!\cdots\!67}{816518130840220}a^{2}+\frac{141949534929547}{204129532710055}a+\frac{118016229704002}{204129532710055}$, $\frac{818879203871}{10\!\cdots\!75}a^{17}-\frac{24560605638083}{816518130840220}a^{16}+\frac{519016367834151}{40\!\cdots\!00}a^{15}-\frac{736349022829787}{40\!\cdots\!00}a^{14}+\frac{14452764878614}{35194747018975}a^{13}-\frac{51\!\cdots\!83}{40\!\cdots\!00}a^{12}+\frac{13\!\cdots\!76}{10\!\cdots\!75}a^{11}-\frac{131441477348877}{163303626168044}a^{10}+\frac{89\!\cdots\!41}{40\!\cdots\!00}a^{9}-\frac{40\!\cdots\!49}{20\!\cdots\!50}a^{8}-\frac{572232444809901}{204129532710055}a^{7}+\frac{77\!\cdots\!47}{40\!\cdots\!00}a^{6}+\frac{19\!\cdots\!34}{10\!\cdots\!75}a^{5}+\frac{19\!\cdots\!69}{10\!\cdots\!75}a^{4}-\frac{441671388662142}{204129532710055}a^{3}-\frac{26\!\cdots\!91}{816518130840220}a^{2}+\frac{208766571012006}{204129532710055}a-\frac{75860343309996}{204129532710055}$ | 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: | \( 265400.055751 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 265400.055751 \cdot 1}{2\cdot\sqrt{2261987535219532470703125}}\cr\approx \mathstrut & 0.857284257172 \end{aligned}\]
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
\(\Q(\sqrt{5}) \), 6.2.4100625.2 x2, 6.2.4100625.1 x2, 9.1.672605015625.1 |
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: | 6.2.4100625.1, 6.2.4100625.2 |
Degree 9 sibling: | 9.1.672605015625.1 |
Degree 12 siblings: | deg 12, deg 12 |
Minimal sibling: | 6.2.4100625.1 |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $18$ | $9$ | $2$ | $32$ | |||
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |