Normalized defining polynomial
\( x^{27} - 18 x^{25} + 189 x^{23} - 63 x^{22} - 1062 x^{21} + 648 x^{20} + 4464 x^{19} - 2679 x^{18} - 14499 x^{17} + 7092 x^{16} + 36963 x^{15} - 21411 x^{14} - 74187 x^{13} + 24255 x^{12} + 61398 x^{11} - 36351 x^{10} - 17520 x^{9} + 66663 x^{8} + 42507 x^{7} - 11583 x^{6} - 18603 x^{5} - 10539 x^{4} - 6327 x^{3} - 2835 x^{2} - 639 x - 53 \)
Invariants
Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[9, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(-17717054310925604811052271173453197606912\)\(\medspace = -\,2^{18}\cdot 3^{73}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $30.95$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $2, 3$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $9$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{20} - \frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{22} - \frac{1}{3} a^{13} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{23} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{24} - \frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{25} - \frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{11635504228846358448927348648104456129976298362080203870293} a^{26} - \frac{243671552971932894966189505516475546636191078626474118814}{3878501409615452816309116216034818709992099454026734623431} a^{25} - \frac{402496355682565522865890882062676263760597275779512889769}{3878501409615452816309116216034818709992099454026734623431} a^{24} - \frac{732113475165636202202006766957359866193410049208005703953}{11635504228846358448927348648104456129976298362080203870293} a^{23} - \frac{444615295729171802380101902949602842609811436545241769002}{3878501409615452816309116216034818709992099454026734623431} a^{22} + \frac{1930138224547042928127337708620447392391031412519046808457}{11635504228846358448927348648104456129976298362080203870293} a^{21} + \frac{63957912991709353558655667472164195740015747186240816000}{3878501409615452816309116216034818709992099454026734623431} a^{20} - \frac{788023945844073307283080035487410611470263178658546990413}{11635504228846358448927348648104456129976298362080203870293} a^{19} + \frac{363312292676128639534226300361200324658971513590440525216}{3878501409615452816309116216034818709992099454026734623431} a^{18} - \frac{214703717668866498046849305967448255473405777892150136337}{11635504228846358448927348648104456129976298362080203870293} a^{17} + \frac{1717224856312352334272441001456070842865724747077479616681}{11635504228846358448927348648104456129976298362080203870293} a^{16} + \frac{109090457486133487227852370635314340738907184827619957501}{11635504228846358448927348648104456129976298362080203870293} a^{15} + \frac{348319907864351190901024064766597369705671888074085652371}{11635504228846358448927348648104456129976298362080203870293} a^{14} - \frac{83052603271863379943808483456993636890209671785399575402}{11635504228846358448927348648104456129976298362080203870293} a^{13} - \frac{5726739346432612054774755763335009788879480465245125545839}{11635504228846358448927348648104456129976298362080203870293} a^{12} - \frac{756324911303443926242673298150075085000843967713200570388}{3878501409615452816309116216034818709992099454026734623431} a^{11} - \frac{5440959265665826375071538496712318639813418141808176087951}{11635504228846358448927348648104456129976298362080203870293} a^{10} - \frac{5765026625579127968539086721555067912153213671321484799814}{11635504228846358448927348648104456129976298362080203870293} a^{9} - \frac{15623389345379607521447892059842740103862704340783960780}{11635504228846358448927348648104456129976298362080203870293} a^{8} - \frac{4124364837579362572532153580576427163443986984052139640083}{11635504228846358448927348648104456129976298362080203870293} a^{7} + \frac{2201277476446152687916791974388944087245088413101960947105}{11635504228846358448927348648104456129976298362080203870293} a^{6} - \frac{2273161887676426629511981478418464549354578657659554141789}{11635504228846358448927348648104456129976298362080203870293} a^{5} + \frac{4471798604251571855348836735035906671680245641860071335351}{11635504228846358448927348648104456129976298362080203870293} a^{4} - \frac{1040228896385672335313638267007509609336761729624314407446}{3878501409615452816309116216034818709992099454026734623431} a^{3} + \frac{700795977678828098278146549292110198156613041581628139018}{3878501409615452816309116216034818709992099454026734623431} a^{2} - \frac{1918515476391826337162876822938636307631771211203646312283}{3878501409615452816309116216034818709992099454026734623431} a + \frac{1510500826948893899066693640611073486249676124872193205569}{11635504228846358448927348648104456129976298362080203870293}$
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);
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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);
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Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 6360080771.530314 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$C_9\times S_3$ (as 27T12):
A solvable group of order 54 |
The 27 conjugacy class representatives for $C_9\times S_3$ |
Character table for $C_9\times S_3$ is not computed |
Intermediate fields
\(\Q(\zeta_{9})^+\), 3.1.972.1, \(\Q(\zeta_{27})^+\), 9.3.74384733888.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 sibling: | data not computed |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | $18{,}\,{\href{/LocalNumberField/5.9.0.1}{9} }$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/LocalNumberField/11.9.0.1}{9} }$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ | $18{,}\,{\href{/LocalNumberField/23.9.0.1}{9} }$ | $18{,}\,{\href{/LocalNumberField/29.9.0.1}{9} }$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ | $18{,}\,{\href{/LocalNumberField/41.9.0.1}{9} }$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/LocalNumberField/47.9.0.1}{9} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{9}$ | $18{,}\,{\href{/LocalNumberField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
2 | Data not computed | ||||||
3 | Data not computed |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.27.9t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.27.9t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.27.9t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
1.27.18t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
1.27.18t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
1.27.18t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
* | 1.27.9t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.27.9t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
1.27.18t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
* | 1.27.9t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
1.27.18t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
1.27.18t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
* | 2.972.3t2.c.a | $2$ | $ 2^{2} \cdot 3^{5}$ | 3.1.972.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.972.6t5.c.a | $2$ | $ 2^{2} \cdot 3^{5}$ | 6.0.2834352.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.972.6t5.c.b | $2$ | $ 2^{2} \cdot 3^{5}$ | 6.0.2834352.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.2916.18t16.c.a | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $C_9\times S_3$ (as 27T12) | $0$ | $0$ |
* | 2.2916.18t16.c.b | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $C_9\times S_3$ (as 27T12) | $0$ | $0$ |
* | 2.2916.18t16.c.c | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $C_9\times S_3$ (as 27T12) | $0$ | $0$ |
* | 2.2916.18t16.c.d | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $C_9\times S_3$ (as 27T12) | $0$ | $0$ |
* | 2.2916.18t16.c.e | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $C_9\times S_3$ (as 27T12) | $0$ | $0$ |
* | 2.2916.18t16.c.f | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $C_9\times S_3$ (as 27T12) | $0$ | $0$ |