Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 14*x^28 - 37*x^27 + 83*x^26 - 152*x^25 + 241*x^24 - 335*x^23 + 465*x^22 - 620*x^21 + 812*x^20 - 1084*x^19 + 1348*x^18 - 1578*x^17 + 1736*x^16 - 1826*x^15 + 1986*x^14 - 2009*x^13 + 2111*x^12 - 2179*x^11 + 1810*x^10 - 1551*x^9 + 1376*x^8 - 959*x^7 + 887*x^6 - 765*x^5 + 462*x^4 - 285*x^3 + 146*x^2 - 13*x + 1)
gp: K = bnfinit(y^30 - 5*y^29 + 14*y^28 - 37*y^27 + 83*y^26 - 152*y^25 + 241*y^24 - 335*y^23 + 465*y^22 - 620*y^21 + 812*y^20 - 1084*y^19 + 1348*y^18 - 1578*y^17 + 1736*y^16 - 1826*y^15 + 1986*y^14 - 2009*y^13 + 2111*y^12 - 2179*y^11 + 1810*y^10 - 1551*y^9 + 1376*y^8 - 959*y^7 + 887*y^6 - 765*y^5 + 462*y^4 - 285*y^3 + 146*y^2 - 13*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 5*x^29 + 14*x^28 - 37*x^27 + 83*x^26 - 152*x^25 + 241*x^24 - 335*x^23 + 465*x^22 - 620*x^21 + 812*x^20 - 1084*x^19 + 1348*x^18 - 1578*x^17 + 1736*x^16 - 1826*x^15 + 1986*x^14 - 2009*x^13 + 2111*x^12 - 2179*x^11 + 1810*x^10 - 1551*x^9 + 1376*x^8 - 959*x^7 + 887*x^6 - 765*x^5 + 462*x^4 - 285*x^3 + 146*x^2 - 13*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 5*x^29 + 14*x^28 - 37*x^27 + 83*x^26 - 152*x^25 + 241*x^24 - 335*x^23 + 465*x^22 - 620*x^21 + 812*x^20 - 1084*x^19 + 1348*x^18 - 1578*x^17 + 1736*x^16 - 1826*x^15 + 1986*x^14 - 2009*x^13 + 2111*x^12 - 2179*x^11 + 1810*x^10 - 1551*x^9 + 1376*x^8 - 959*x^7 + 887*x^6 - 765*x^5 + 462*x^4 - 285*x^3 + 146*x^2 - 13*x + 1)
\( x^{30} - 5 x^{29} + 14 x^{28} - 37 x^{27} + 83 x^{26} - 152 x^{25} + 241 x^{24} - 335 x^{23} + 465 x^{22} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-141684422447224323645012862864040611366932987\)
\(\medspace = -\,3^{15}\cdot 439^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.63\) | 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: | $3^{1/2}439^{1/2}\approx 36.29049462324811$ | ||
| Ramified primes: |
\(3\), \(439\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{8}+\frac{1}{3}$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{9}+\frac{1}{3}a$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{10}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{17}+\frac{1}{9}a^{16}+\frac{1}{3}a^{15}-\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{9}a^{11}-\frac{2}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{9}a^{3}+\frac{4}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{18}+\frac{1}{9}a^{17}-\frac{1}{3}a^{15}-\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{9}a^{12}-\frac{2}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{9}a^{4}+\frac{4}{9}a^{2}+\frac{1}{9}a-\frac{1}{3}$, $\frac{1}{9}a^{21}+\frac{1}{9}a^{18}-\frac{1}{9}a^{17}-\frac{1}{9}a^{16}+\frac{1}{3}a^{15}+\frac{4}{9}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{9}a^{10}-\frac{4}{9}a^{9}-\frac{4}{9}a^{8}-\frac{1}{3}a^{7}-\frac{2}{9}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{9}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{27}a^{22}-\frac{1}{27}a^{20}-\frac{1}{27}a^{19}+\frac{4}{27}a^{18}+\frac{2}{27}a^{17}-\frac{2}{27}a^{16}+\frac{2}{9}a^{15}+\frac{13}{27}a^{14}+\frac{4}{9}a^{13}+\frac{2}{27}a^{12}+\frac{8}{27}a^{11}+\frac{4}{27}a^{10}-\frac{4}{27}a^{9}-\frac{11}{27}a^{8}-\frac{2}{9}a^{7}+\frac{7}{27}a^{6}-\frac{4}{9}a^{5}-\frac{13}{27}a^{4}-\frac{10}{27}a^{3}-\frac{5}{27}a^{2}-\frac{1}{27}a-\frac{11}{27}$, $\frac{1}{27}a^{23}-\frac{1}{27}a^{21}-\frac{1}{27}a^{20}+\frac{1}{27}a^{19}+\frac{2}{27}a^{18}+\frac{4}{27}a^{17}+\frac{1}{9}a^{16}+\frac{4}{27}a^{15}-\frac{2}{9}a^{14}+\frac{11}{27}a^{13}-\frac{10}{27}a^{12}+\frac{1}{27}a^{11}-\frac{4}{27}a^{10}+\frac{4}{27}a^{9}-\frac{1}{3}a^{8}-\frac{11}{27}a^{7}+\frac{2}{9}a^{6}+\frac{5}{27}a^{5}+\frac{8}{27}a^{4}-\frac{8}{27}a^{3}-\frac{1}{27}a^{2}+\frac{13}{27}a-\frac{1}{9}$, $\frac{1}{27}a^{24}-\frac{1}{27}a^{21}+\frac{1}{27}a^{19}-\frac{1}{27}a^{18}-\frac{4}{27}a^{17}+\frac{2}{27}a^{16}-\frac{1}{9}a^{14}+\frac{2}{27}a^{13}+\frac{1}{9}a^{12}+\frac{4}{27}a^{11}-\frac{1}{27}a^{10}+\frac{5}{27}a^{9}+\frac{5}{27}a^{8}+\frac{4}{9}a^{6}-\frac{4}{27}a^{5}+\frac{2}{9}a^{4}-\frac{11}{27}a^{3}-\frac{1}{27}a^{2}-\frac{13}{27}a-\frac{11}{27}$, $\frac{1}{459}a^{25}-\frac{1}{153}a^{24}+\frac{7}{459}a^{23}-\frac{4}{459}a^{22}-\frac{1}{459}a^{21}+\frac{2}{51}a^{20}+\frac{1}{153}a^{19}-\frac{56}{459}a^{18}+\frac{8}{153}a^{17}-\frac{19}{153}a^{16}+\frac{133}{459}a^{15}+\frac{200}{459}a^{14}-\frac{4}{459}a^{13}+\frac{49}{153}a^{12}+\frac{40}{153}a^{11}-\frac{179}{459}a^{10}+\frac{5}{51}a^{9}+\frac{49}{153}a^{8}-\frac{92}{459}a^{7}+\frac{143}{459}a^{6}-\frac{133}{459}a^{5}+\frac{4}{9}a^{4}+\frac{31}{153}a^{3}+\frac{112}{459}a^{2}+\frac{56}{459}a-\frac{38}{153}$, $\frac{1}{459}a^{26}-\frac{2}{459}a^{24}+\frac{4}{459}a^{22}-\frac{19}{459}a^{21}+\frac{2}{153}a^{20}+\frac{7}{153}a^{19}-\frac{59}{459}a^{18}-\frac{70}{459}a^{17}+\frac{10}{153}a^{16}+\frac{7}{153}a^{15}-\frac{152}{459}a^{14}-\frac{205}{459}a^{13}-\frac{4}{9}a^{12}+\frac{32}{153}a^{11}-\frac{152}{459}a^{10}-\frac{58}{459}a^{9}+\frac{1}{51}a^{8}+\frac{35}{153}a^{7}+\frac{7}{459}a^{6}-\frac{229}{459}a^{5}-\frac{6}{17}a^{4}-\frac{1}{3}a^{3}+\frac{74}{153}a^{2}-\frac{82}{459}a+\frac{134}{459}$, $\frac{1}{18819}a^{27}-\frac{4}{18819}a^{26}+\frac{11}{18819}a^{25}-\frac{65}{18819}a^{24}+\frac{37}{2091}a^{23}+\frac{100}{18819}a^{22}+\frac{40}{6273}a^{21}+\frac{571}{18819}a^{20}+\frac{107}{6273}a^{19}+\frac{317}{6273}a^{18}-\frac{517}{18819}a^{17}+\frac{1727}{18819}a^{16}+\frac{322}{6273}a^{15}+\frac{3343}{18819}a^{14}-\frac{271}{6273}a^{13}-\frac{2192}{18819}a^{12}-\frac{3070}{6273}a^{11}-\frac{137}{2091}a^{10}-\frac{35}{459}a^{9}+\frac{6026}{18819}a^{8}+\frac{1991}{6273}a^{7}-\frac{4025}{18819}a^{6}-\frac{998}{2091}a^{5}-\frac{4163}{18819}a^{4}-\frac{206}{459}a^{3}-\frac{7283}{18819}a^{2}+\frac{14}{369}a-\frac{8699}{18819}$, $\frac{1}{90952227}a^{28}-\frac{50}{3368601}a^{27}-\frac{3953}{90952227}a^{26}-\frac{58331}{90952227}a^{25}+\frac{47369}{30317409}a^{24}-\frac{172267}{10105803}a^{23}+\frac{10163}{2218347}a^{22}-\frac{1638122}{30317409}a^{21}-\frac{4869967}{90952227}a^{20}+\frac{93700}{2218347}a^{19}+\frac{7303033}{90952227}a^{18}+\frac{10881722}{90952227}a^{17}+\frac{5518207}{90952227}a^{16}-\frac{10157443}{30317409}a^{15}-\frac{28940099}{90952227}a^{14}+\frac{9400367}{30317409}a^{13}+\frac{32771510}{90952227}a^{12}+\frac{29485949}{90952227}a^{11}+\frac{18353260}{90952227}a^{10}-\frac{22248364}{90952227}a^{9}-\frac{37284641}{90952227}a^{8}+\frac{9632029}{30317409}a^{7}+\frac{5991811}{90952227}a^{6}-\frac{102539}{1122867}a^{5}+\frac{14884363}{90952227}a^{4}+\frac{26116646}{90952227}a^{3}-\frac{15092483}{30317409}a^{2}+\frac{33734584}{90952227}a+\frac{8100526}{90952227}$, $\frac{1}{8718043814631}a^{29}-\frac{36127}{8718043814631}a^{28}+\frac{106912861}{8718043814631}a^{27}-\frac{21108641}{35876723517}a^{26}-\frac{3015072001}{8718043814631}a^{25}+\frac{21357217930}{2906014604877}a^{24}+\frac{56018246680}{8718043814631}a^{23}-\frac{124499265928}{8718043814631}a^{22}+\frac{169712741441}{8718043814631}a^{21}-\frac{29141350688}{968671534959}a^{20}+\frac{334184994851}{8718043814631}a^{19}-\frac{381887125712}{8718043814631}a^{18}-\frac{5135614850}{212635214991}a^{17}-\frac{430484801305}{8718043814631}a^{16}-\frac{592793941340}{8718043814631}a^{15}+\frac{1666832596355}{8718043814631}a^{14}-\frac{4044430113829}{8718043814631}a^{13}+\frac{900944279843}{2906014604877}a^{12}+\frac{889790626832}{8718043814631}a^{11}-\frac{1312745134775}{8718043814631}a^{10}-\frac{2807199929740}{8718043814631}a^{9}-\frac{405632310460}{8718043814631}a^{8}+\frac{36181679936}{512826106743}a^{7}-\frac{2531012214940}{8718043814631}a^{6}-\frac{102640181165}{8718043814631}a^{5}-\frac{113379010646}{8718043814631}a^{4}+\frac{3184122242}{97955548479}a^{3}-\frac{1066973953016}{8718043814631}a^{2}-\frac{135726440162}{2906014604877}a+\frac{1521117204008}{8718043814631}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{10493915024}{107630170551} a^{29} + \frac{49355026628}{107630170551} a^{28} - \frac{132614083868}{107630170551} a^{27} + \frac{117342677845}{35876723517} a^{26} - \frac{778750507993}{107630170551} a^{25} + \frac{465876170536}{35876723517} a^{24} - \frac{2195423870762}{107630170551} a^{23} + \frac{3033012597578}{107630170551} a^{22} - \frac{251334944627}{6331186503} a^{21} + \frac{1889209152911}{35876723517} a^{20} - \frac{7406617182097}{107630170551} a^{19} + \frac{9927122893516}{107630170551} a^{18} - \frac{12194172448426}{107630170551} a^{17} + \frac{14243506409576}{107630170551} a^{16} - \frac{921233826571}{6331186503} a^{15} + \frac{16551488080124}{107630170551} a^{14} - \frac{18182842047562}{107630170551} a^{13} + \frac{6001703474113}{35876723517} a^{12} - \frac{19196833721899}{107630170551} a^{11} + \frac{1158491948621}{6331186503} a^{10} - \frac{15767986812016}{107630170551} a^{9} + \frac{14195308636277}{107630170551} a^{8} - \frac{12730208819687}{107630170551} a^{7} + \frac{8286655239413}{107630170551} a^{6} - \frac{8144006299112}{107630170551} a^{5} + \frac{6821780067499}{107630170551} a^{4} - \frac{43866346123}{1209327759} a^{3} + \frac{2551977980950}{107630170551} a^{2} - \frac{469868085199}{35876723517} a + \frac{3070603741}{2625126111} \)
(order $6$)
| 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: |
$\frac{11334416479}{8718043814631}a^{29}+\frac{97377215960}{8718043814631}a^{28}-\frac{758160617063}{8718043814631}a^{27}+\frac{27262156136}{107630170551}a^{26}-\frac{5553370812400}{8718043814631}a^{25}+\frac{4373552534833}{2906014604877}a^{24}-\frac{24151146028166}{8718043814631}a^{23}+\frac{36669539197112}{8718043814631}a^{22}-\frac{1180053006182}{212635214991}a^{21}+\frac{7051979073424}{968671534959}a^{20}-\frac{88833418107283}{8718043814631}a^{19}+\frac{114081532329427}{8718043814631}a^{18}-\frac{151739108112634}{8718043814631}a^{17}+\frac{192021339531662}{8718043814631}a^{16}-\frac{210313155569846}{8718043814631}a^{15}+\frac{224034177829964}{8718043814631}a^{14}-\frac{223923188307892}{8718043814631}a^{13}+\frac{82380488311637}{2906014604877}a^{12}-\frac{266810415947218}{8718043814631}a^{11}+\frac{249473739621907}{8718043814631}a^{10}-\frac{282104764223239}{8718043814631}a^{9}+\frac{212695065041576}{8718043814631}a^{8}-\frac{7140379724983}{512826106743}a^{7}+\frac{163584713181002}{8718043814631}a^{6}-\frac{115381956984140}{8718043814631}a^{5}+\frac{72212172916411}{8718043814631}a^{4}-\frac{1682996482}{140538807}a^{3}+\frac{35377607944840}{8718043814631}a^{2}-\frac{1058440161275}{2906014604877}a+\frac{8430372114857}{8718043814631}$, $\frac{16739126125}{8718043814631}a^{29}-\frac{72607634881}{8718043814631}a^{28}+\frac{220123532803}{8718043814631}a^{27}-\frac{7760076466}{107630170551}a^{26}+\frac{1369787926790}{8718043814631}a^{25}-\frac{880331324318}{2906014604877}a^{24}+\frac{261207493958}{512826106743}a^{23}-\frac{6381942043783}{8718043814631}a^{22}+\frac{8950585753724}{8718043814631}a^{21}-\frac{1270310139185}{968671534959}a^{20}+\frac{16035651985220}{8718043814631}a^{19}-\frac{20949828524723}{8718043814631}a^{18}+\frac{1555620609769}{512826106743}a^{17}-\frac{31841544966391}{8718043814631}a^{16}+\frac{35259364955866}{8718043814631}a^{15}-\frac{36790210242499}{8718043814631}a^{14}+\frac{37857351417629}{8718043814631}a^{13}-\frac{13365960181588}{2906014604877}a^{12}+\frac{1087918930942}{212635214991}a^{11}-\frac{41078493669689}{8718043814631}a^{10}+\frac{38050613863676}{8718043814631}a^{9}-\frac{31328272159276}{8718043814631}a^{8}+\frac{21284237140516}{8718043814631}a^{7}-\frac{8286424138237}{8718043814631}a^{6}+\frac{18620519813701}{8718043814631}a^{5}-\frac{12255866402237}{8718043814631}a^{4}-\frac{1834944196}{97955548479}a^{3}-\frac{4384221118673}{8718043814631}a^{2}+\frac{130434040738}{2906014604877}a+\frac{12198148944443}{8718043814631}$, $\frac{229259756290}{2906014604877}a^{29}-\frac{1115372670439}{2906014604877}a^{28}+\frac{3094627902562}{2906014604877}a^{27}-\frac{53167387850}{18993559509}a^{26}+\frac{17905977560606}{2906014604877}a^{25}-\frac{10787466429992}{968671534959}a^{24}+\frac{50278687842919}{2906014604877}a^{23}-\frac{67623050619019}{2906014604877}a^{22}+\frac{92408689328588}{2906014604877}a^{21}-\frac{13452531369070}{322890511653}a^{20}+\frac{159361642595396}{2906014604877}a^{19}-\frac{212040439571021}{2906014604877}a^{18}+\frac{259873834759079}{2906014604877}a^{17}-\frac{301226719329970}{2906014604877}a^{16}+\frac{319561749460171}{2906014604877}a^{15}-\frac{19293307515899}{170942035581}a^{14}+\frac{357060195725201}{2906014604877}a^{13}-\frac{118419465218620}{968671534959}a^{12}+\frac{382344316525658}{2906014604877}a^{11}-\frac{380179723927871}{2906014604877}a^{10}+\frac{300961099622801}{2906014604877}a^{9}-\frac{250241925088567}{2906014604877}a^{8}+\frac{198832465120372}{2906014604877}a^{7}-\frac{147671080677199}{2906014604877}a^{6}+\frac{8929586880725}{170942035581}a^{5}-\frac{108392029929398}{2906014604877}a^{4}+\frac{831525260237}{32651849493}a^{3}-\frac{30054264448472}{2906014604877}a^{2}+\frac{52267317476}{56980678527}a+\frac{3371491688744}{2906014604877}$, $\frac{4568312374}{2906014604877}a^{29}+\frac{11675020322}{2906014604877}a^{28}-\frac{127176194921}{2906014604877}a^{27}+\frac{39525299963}{322890511653}a^{26}-\frac{921313780180}{2906014604877}a^{25}+\frac{18261011693}{23626134999}a^{24}-\frac{4105379425988}{2906014604877}a^{23}+\frac{6193590303299}{2906014604877}a^{22}-\frac{195716029274}{70878404997}a^{21}+\frac{1185707959535}{322890511653}a^{20}-\frac{14951233736398}{2906014604877}a^{19}+\frac{18861097537222}{2906014604877}a^{18}-\frac{25706790679714}{2906014604877}a^{17}+\frac{32747744813057}{2906014604877}a^{16}-\frac{35648569512959}{2906014604877}a^{15}+\frac{37970847136940}{2906014604877}a^{14}-\frac{38053835110759}{2906014604877}a^{13}+\frac{14536514389520}{968671534959}a^{12}-\frac{45570856692475}{2906014604877}a^{11}+\frac{42681353392288}{2906014604877}a^{10}-\frac{50126211672913}{2906014604877}a^{9}+\frac{36692205079955}{2906014604877}a^{8}-\frac{20553574580981}{2906014604877}a^{7}+\frac{32491606830845}{2906014604877}a^{6}-\frac{19863895286279}{2906014604877}a^{5}+\frac{12382082413687}{2906014604877}a^{4}-\frac{224520527506}{32651849493}a^{3}+\frac{6239938593127}{2906014604877}a^{2}-\frac{186767063939}{968671534959}a+\frac{2854026145511}{2906014604877}$, $\frac{3114548492}{107630170551}a^{29}-\frac{841219204}{6331186503}a^{28}+\frac{36246822353}{107630170551}a^{27}-\frac{10249382111}{11958907839}a^{26}+\frac{197446572040}{107630170551}a^{25}-\frac{111203216674}{35876723517}a^{24}+\frac{482448935462}{107630170551}a^{23}-\frac{606976427501}{107630170551}a^{22}+\frac{838599856147}{107630170551}a^{21}-\frac{7283933447}{703465167}a^{20}+\frac{1448280992500}{107630170551}a^{19}-\frac{1951625003926}{107630170551}a^{18}+\frac{2315891498296}{107630170551}a^{17}-\frac{2556461429957}{107630170551}a^{16}+\frac{2610400753700}{107630170551}a^{15}-\frac{2658073190102}{107630170551}a^{14}+\frac{3077164860877}{107630170551}a^{13}-\frac{985273631255}{35876723517}a^{12}+\frac{3171877710739}{107630170551}a^{11}-\frac{3225999381424}{107630170551}a^{10}+\frac{2080753565947}{107630170551}a^{9}-\frac{1709418253337}{107630170551}a^{8}+\frac{1777009268603}{107630170551}a^{7}-\frac{1164096327176}{107630170551}a^{6}+\frac{1206064925861}{107630170551}a^{5}-\frac{944083936141}{107630170551}a^{4}+\frac{4930200010}{1209327759}a^{3}-\frac{122507337907}{107630170551}a^{2}+\frac{3508713977}{35876723517}a-\frac{10493915024}{107630170551}$, $\frac{3623814187}{512826106743}a^{29}-\frac{471905884046}{8718043814631}a^{28}+\frac{2503936997}{12507953823}a^{27}-\frac{171519315998}{322890511653}a^{26}+\frac{10880262184444}{8718043814631}a^{25}-\frac{7433353487467}{2906014604877}a^{24}+\frac{2195995887871}{512826106743}a^{23}-\frac{53345430844214}{8718043814631}a^{22}+\frac{70958116175935}{8718043814631}a^{21}-\frac{10578187511089}{968671534959}a^{20}+\frac{130015550274940}{8718043814631}a^{19}-\frac{168284731551523}{8718043814631}a^{18}+\frac{218224029454054}{8718043814631}a^{17}-\frac{15513932689867}{512826106743}a^{16}+\frac{285986356024181}{8718043814631}a^{15}-\frac{299896135059956}{8718043814631}a^{14}+\frac{7636536800126}{212635214991}a^{13}-\frac{114305983617221}{2906014604877}a^{12}+\frac{356121313955362}{8718043814631}a^{11}-\frac{344219637489346}{8718043814631}a^{10}+\frac{344302987951234}{8718043814631}a^{9}-\frac{254641383988223}{8718043814631}a^{8}+\frac{176710178803166}{8718043814631}a^{7}-\frac{194405237026169}{8718043814631}a^{6}+\frac{148694378717702}{8718043814631}a^{5}-\frac{100733049094933}{8718043814631}a^{4}+\frac{1116796162315}{97955548479}a^{3}-\frac{35938684527583}{8718043814631}a^{2}+\frac{1069045795772}{2906014604877}a-\frac{3352607279900}{8718043814631}$, $\frac{525133364107}{8718043814631}a^{29}-\frac{2711104200706}{8718043814631}a^{28}+\frac{7718621686615}{8718043814631}a^{27}-\frac{246616446068}{107630170551}a^{26}+\frac{44372592466547}{8718043814631}a^{25}-\frac{27021553168241}{2906014604877}a^{24}+\frac{125522935832011}{8718043814631}a^{23}-\frac{168433829943616}{8718043814631}a^{22}+\frac{228231594997379}{8718043814631}a^{21}-\frac{33834714199010}{968671534959}a^{20}+\frac{403443925703669}{8718043814631}a^{19}-\frac{533702155834778}{8718043814631}a^{18}+\frac{660964649922788}{8718043814631}a^{17}-\frac{764451611880181}{8718043814631}a^{16}+\frac{809889183573103}{8718043814631}a^{15}-\frac{834862466750563}{8718043814631}a^{14}+\frac{912962248327844}{8718043814631}a^{13}-\frac{312607569327406}{2906014604877}a^{12}+\frac{984000388272005}{8718043814631}a^{11}-\frac{979852170524789}{8718043814631}a^{10}+\frac{807254023331981}{8718043814631}a^{9}-\frac{629179469439103}{8718043814631}a^{8}+\frac{519432600943195}{8718043814631}a^{7}-\frac{25967596351166}{512826106743}a^{6}+\frac{392720865062161}{8718043814631}a^{5}-\frac{284464922494277}{8718043814631}a^{4}+\frac{2279872199348}{97955548479}a^{3}-\frac{71978107796018}{8718043814631}a^{2}+\frac{2122406932264}{2906014604877}a-\frac{270414133970}{212635214991}$, $\frac{1201920345997}{8718043814631}a^{29}-\frac{5544065519791}{8718043814631}a^{28}+\frac{14577212227054}{8718043814631}a^{27}-\frac{1426489785575}{322890511653}a^{26}+\frac{84401036072255}{8718043814631}a^{25}-\frac{49639687842122}{2906014604877}a^{24}+\frac{230444511194032}{8718043814631}a^{23}-\frac{313073292918559}{8718043814631}a^{22}+\frac{440020191437555}{8718043814631}a^{21}-\frac{64579655868092}{968671534959}a^{20}+\frac{756531537111971}{8718043814631}a^{19}-\frac{10\!\cdots\!49}{8718043814631}a^{18}+\frac{72753233553889}{512826106743}a^{17}-\frac{14\!\cdots\!85}{8718043814631}a^{16}+\frac{91615433003951}{512826106743}a^{15}-\frac{16\!\cdots\!69}{8718043814631}a^{14}+\frac{18\!\cdots\!41}{8718043814631}a^{13}-\frac{586622332748503}{2906014604877}a^{12}+\frac{18\!\cdots\!81}{8718043814631}a^{11}-\frac{19\!\cdots\!14}{8718043814631}a^{10}+\frac{14\!\cdots\!75}{8718043814631}a^{9}-\frac{13\!\cdots\!45}{8718043814631}a^{8}+\frac{12\!\cdots\!13}{8718043814631}a^{7}-\frac{722525664032446}{8718043814631}a^{6}+\frac{783491625201355}{8718043814631}a^{5}-\frac{648379454204276}{8718043814631}a^{4}+\frac{3461032648796}{97955548479}a^{3}-\frac{221690791479461}{8718043814631}a^{2}+\frac{38492777661247}{2906014604877}a+\frac{20508720002603}{8718043814631}$, $\frac{2039003846443}{8718043814631}a^{29}-\frac{9922904408344}{8718043814631}a^{28}+\frac{663266111819}{212635214991}a^{27}-\frac{23911221351700}{2906014604877}a^{26}+\frac{159578014773506}{8718043814631}a^{25}-\frac{96158407625525}{2906014604877}a^{24}+\frac{452862568859995}{8718043814631}a^{23}-\frac{623515374939250}{8718043814631}a^{22}+\frac{867694995827714}{8718043814631}a^{21}-\frac{384610929002783}{2906014604877}a^{20}+\frac{15\!\cdots\!12}{8718043814631}a^{19}-\frac{20\!\cdots\!42}{8718043814631}a^{18}+\frac{24\!\cdots\!57}{8718043814631}a^{17}-\frac{28\!\cdots\!05}{8718043814631}a^{16}+\frac{31\!\cdots\!87}{8718043814631}a^{15}-\frac{33\!\cdots\!61}{8718043814631}a^{14}+\frac{36\!\cdots\!41}{8718043814631}a^{13}-\frac{134993235614483}{322890511653}a^{12}+\frac{226293181439698}{512826106743}a^{11}-\frac{39\!\cdots\!95}{8718043814631}a^{10}+\frac{31\!\cdots\!82}{8718043814631}a^{9}-\frac{27\!\cdots\!37}{8718043814631}a^{8}+\frac{24\!\cdots\!66}{8718043814631}a^{7}-\frac{16\!\cdots\!83}{8718043814631}a^{6}+\frac{15\!\cdots\!84}{8718043814631}a^{5}-\frac{13\!\cdots\!18}{8718043814631}a^{4}+\frac{8527563025343}{97955548479}a^{3}-\frac{469726504898789}{8718043814631}a^{2}+\frac{8731487817236}{322890511653}a+\frac{10735127033645}{8718043814631}$, $\frac{326595400}{6331186503}a^{29}-\frac{25323035692}{107630170551}a^{28}+\frac{66707395976}{107630170551}a^{27}-\frac{174967932977}{107630170551}a^{26}+\frac{378102615422}{107630170551}a^{25}-\frac{221088823406}{35876723517}a^{24}+\frac{1008404533604}{107630170551}a^{23}-\frac{1334005204858}{107630170551}a^{22}+\frac{1854707859643}{107630170551}a^{21}-\frac{2426207512504}{107630170551}a^{20}+\frac{355173022943}{11958907839}a^{19}-\frac{1423407939071}{35876723517}a^{18}+\frac{1721887124371}{35876723517}a^{17}-\frac{5911478325583}{107630170551}a^{16}+\frac{6283466549174}{107630170551}a^{15}-\frac{157685575886}{2625126111}a^{14}+\frac{424675204328}{6331186503}a^{13}-\frac{7011899083501}{107630170551}a^{12}+\frac{851605658896}{11958907839}a^{11}-\frac{845055575875}{11958907839}a^{10}+\frac{37057775533}{703465167}a^{9}-\frac{5022254821198}{107630170551}a^{8}+\frac{4326883028939}{107630170551}a^{7}-\frac{2774611045864}{107630170551}a^{6}+\frac{3314217317912}{107630170551}a^{5}-\frac{713073594983}{35876723517}a^{4}+\frac{15392622728}{1209327759}a^{3}-\frac{651199091098}{107630170551}a^{2}+\frac{281918029609}{107630170551}a+\frac{91584133727}{107630170551}$, $\frac{464137909906}{8718043814631}a^{29}-\frac{2760579432625}{8718043814631}a^{28}+\frac{8453393584012}{8718043814631}a^{27}-\frac{2385702033883}{968671534959}a^{26}+\frac{48623122909103}{8718043814631}a^{25}-\frac{30525583113137}{2906014604877}a^{24}+\frac{142364038522864}{8718043814631}a^{23}-\frac{190889749614169}{8718043814631}a^{22}+\frac{251485123411007}{8718043814631}a^{21}-\frac{37758736560325}{968671534959}a^{20}+\frac{453503244807020}{8718043814631}a^{19}-\frac{590784748045232}{8718043814631}a^{18}+\frac{18212156514082}{212635214991}a^{17}-\frac{50660878829915}{512826106743}a^{16}+\frac{900566721429580}{8718043814631}a^{15}-\frac{22393319854232}{212635214991}a^{14}+\frac{985859236584044}{8718043814631}a^{13}-\frac{352548144390181}{2906014604877}a^{12}+\frac{10\!\cdots\!06}{8718043814631}a^{11}-\frac{10\!\cdots\!95}{8718043814631}a^{10}+\frac{946004353796801}{8718043814631}a^{9}-\frac{627633984139975}{8718043814631}a^{8}+\frac{505532288458978}{8718043814631}a^{7}-\frac{534077695274833}{8718043814631}a^{6}+\frac{388212054846286}{8718043814631}a^{5}-\frac{292390374188054}{8718043814631}a^{4}+\frac{2765220983081}{97955548479}a^{3}-\frac{55446372106541}{8718043814631}a^{2}-\frac{1379129375642}{2906014604877}a+\frac{2801107359173}{8718043814631}$, $\frac{123705576466}{968671534959}a^{29}-\frac{1727300889704}{2906014604877}a^{28}+\frac{1531216889914}{968671534959}a^{27}-\frac{12166295305430}{2906014604877}a^{26}+\frac{26720449152556}{2906014604877}a^{25}-\frac{930441080759}{56980678527}a^{24}+\frac{24616451996185}{968671534959}a^{23}-\frac{100589857557086}{2906014604877}a^{22}+\frac{193161440705}{3986302613}a^{21}-\frac{185877483362923}{2906014604877}a^{20}+\frac{242570884534103}{2906014604877}a^{19}-\frac{326567482138265}{2906014604877}a^{18}+\frac{397195973369423}{2906014604877}a^{17}-\frac{460602241214546}{2906014604877}a^{16}+\frac{55759348184312}{322890511653}a^{15}-\frac{523495964842856}{2906014604877}a^{14}+\frac{192398857717147}{968671534959}a^{13}-\frac{566407605266326}{2906014604877}a^{12}+\frac{607581780544523}{2906014604877}a^{11}-\frac{623186376562031}{2906014604877}a^{10}+\frac{473135827180835}{2906014604877}a^{9}-\frac{432890444159570}{2906014604877}a^{8}+\frac{127404362119298}{968671534959}a^{7}-\frac{231242915207021}{2906014604877}a^{6}+\frac{86200274883787}{968671534959}a^{5}-\frac{202696953975239}{2906014604877}a^{4}+\frac{1121216444713}{32651849493}a^{3}-\frac{26124527991880}{968671534959}a^{2}+\frac{24849084801970}{2906014604877}a+\frac{1153472271859}{2906014604877}$, $\frac{304764147533}{2906014604877}a^{29}-\frac{1407684091651}{2906014604877}a^{28}+\frac{3685651818620}{2906014604877}a^{27}-\frac{9665973911114}{2906014604877}a^{26}+\frac{21116952333323}{2906014604877}a^{25}-\frac{12358170054047}{968671534959}a^{24}+\frac{56784891490814}{2906014604877}a^{23}-\frac{1864610677370}{70878404997}a^{22}+\frac{107402663246956}{2906014604877}a^{21}-\frac{142346893808851}{2906014604877}a^{20}+\frac{61952581479325}{968671534959}a^{19}-\frac{83338665160057}{968671534959}a^{18}+\frac{101488849736807}{968671534959}a^{17}-\frac{350357083869487}{2906014604877}a^{16}+\frac{379416332269577}{2906014604877}a^{15}-\frac{397893455268919}{2906014604877}a^{14}+\frac{442912885683778}{2906014604877}a^{13}-\frac{436028944566025}{2906014604877}a^{12}+\frac{3795456300674}{23626134999}a^{11}-\frac{159964080047702}{968671534959}a^{10}+\frac{121656644058118}{968671534959}a^{9}-\frac{325833700852243}{2906014604877}a^{8}+\frac{304180284090665}{2906014604877}a^{7}-\frac{191725917849535}{2906014604877}a^{6}+\frac{198105969854090}{2906014604877}a^{5}-\frac{55396676773358}{968671534959}a^{4}+\frac{959237893541}{32651849493}a^{3}-\frac{55111243122667}{2906014604877}a^{2}+\frac{30806361997552}{2906014604877}a+\frac{493774418609}{2906014604877}$, $\frac{644047590535}{8718043814631}a^{29}-\frac{3458156626177}{8718043814631}a^{28}+\frac{10361417720926}{8718043814631}a^{27}-\frac{3102683800378}{968671534959}a^{26}+\frac{64196139485306}{8718043814631}a^{25}-\frac{40778299496717}{2906014604877}a^{24}+\frac{201508840829005}{8718043814631}a^{23}-\frac{290023182463477}{8718043814631}a^{22}+\frac{402255247342748}{8718043814631}a^{21}-\frac{59681761750102}{968671534959}a^{20}+\frac{706729549010705}{8718043814631}a^{19}-\frac{940351053418616}{8718043814631}a^{18}+\frac{11\!\cdots\!64}{8718043814631}a^{17}-\frac{83053142175905}{512826106743}a^{16}+\frac{15\!\cdots\!41}{8718043814631}a^{15}-\frac{16\!\cdots\!10}{8718043814631}a^{14}+\frac{18\!\cdots\!39}{8718043814631}a^{13}-\frac{615145594121449}{2906014604877}a^{12}+\frac{19\!\cdots\!94}{8718043814631}a^{11}-\frac{19\!\cdots\!07}{8718043814631}a^{10}+\frac{17\!\cdots\!95}{8718043814631}a^{9}-\frac{15\!\cdots\!06}{8718043814631}a^{8}+\frac{12\!\cdots\!30}{8718043814631}a^{7}-\frac{957038907506854}{8718043814631}a^{6}+\frac{830446319056723}{8718043814631}a^{5}-\frac{725289039337526}{8718043814631}a^{4}+\frac{5464872447197}{97955548479}a^{3}-\frac{298492274046668}{8718043814631}a^{2}+\frac{51742853983717}{2906014604877}a-\frac{33565641585382}{8718043814631}$
| 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: | \( 18103283853.50697 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 18103283853.50697 \cdot 2}{6\cdot\sqrt{141684422447224323645012862864040611366932987}}\cr\approx \mathstrut & 0.476072482619609 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 14*x^28 - 37*x^27 + 83*x^26 - 152*x^25 + 241*x^24 - 335*x^23 + 465*x^22 - 620*x^21 + 812*x^20 - 1084*x^19 + 1348*x^18 - 1578*x^17 + 1736*x^16 - 1826*x^15 + 1986*x^14 - 2009*x^13 + 2111*x^12 - 2179*x^11 + 1810*x^10 - 1551*x^9 + 1376*x^8 - 959*x^7 + 887*x^6 - 765*x^5 + 462*x^4 - 285*x^3 + 146*x^2 - 13*x + 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^30 - 5*x^29 + 14*x^28 - 37*x^27 + 83*x^26 - 152*x^25 + 241*x^24 - 335*x^23 + 465*x^22 - 620*x^21 + 812*x^20 - 1084*x^19 + 1348*x^18 - 1578*x^17 + 1736*x^16 - 1826*x^15 + 1986*x^14 - 2009*x^13 + 2111*x^12 - 2179*x^11 + 1810*x^10 - 1551*x^9 + 1376*x^8 - 959*x^7 + 887*x^6 - 765*x^5 + 462*x^4 - 285*x^3 + 146*x^2 - 13*x + 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^30 - 5*x^29 + 14*x^28 - 37*x^27 + 83*x^26 - 152*x^25 + 241*x^24 - 335*x^23 + 465*x^22 - 620*x^21 + 812*x^20 - 1084*x^19 + 1348*x^18 - 1578*x^17 + 1736*x^16 - 1826*x^15 + 1986*x^14 - 2009*x^13 + 2111*x^12 - 2179*x^11 + 1810*x^10 - 1551*x^9 + 1376*x^8 - 959*x^7 + 887*x^6 - 765*x^5 + 462*x^4 - 285*x^3 + 146*x^2 - 13*x + 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 5*x^29 + 14*x^28 - 37*x^27 + 83*x^26 - 152*x^25 + 241*x^24 - 335*x^23 + 465*x^22 - 620*x^21 + 812*x^20 - 1084*x^19 + 1348*x^18 - 1578*x^17 + 1736*x^16 - 1826*x^15 + 1986*x^14 - 2009*x^13 + 2111*x^12 - 2179*x^11 + 1810*x^10 - 1551*x^9 + 1376*x^8 - 959*x^7 + 887*x^6 - 765*x^5 + 462*x^4 - 285*x^3 + 146*x^2 - 13*x + 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 60 |
| The 18 conjugacy class representatives for $D_{30}$ |
| Character table for $D_{30}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.439.1, 5.1.192721.1, 6.0.5203467.1, 10.0.9025356273363.1, 15.1.3142328914862177479.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 30 sibling: | deg 30 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $30$ | R | ${\href{/padicField/5.10.0.1}{10} }^{3}$ | $15^{2}$ | $30$ | ${\href{/padicField/13.3.0.1}{3} }^{10}$ | ${\href{/padicField/17.2.0.1}{2} }^{15}$ | ${\href{/padicField/19.5.0.1}{5} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{15}$ | $30$ | ${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{14}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{15}$ | ${\href{/padicField/43.2.0.1}{2} }^{14}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{15}$ | $30$ | ${\href{/padicField/59.2.0.1}{2} }^{15}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(439\)
| $\Q_{439}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{439}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.1317.2t1.a.a | $1$ | $ 3 \cdot 439 $ | \(\Q(\sqrt{1317}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.439.2t1.a.a | $1$ | $ 439 $ | \(\Q(\sqrt{-439}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 2.3951.6t3.b.a | $2$ | $ 3^{2} \cdot 439 $ | 6.2.2284322013.1 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.439.3t2.a.a | $2$ | $ 439 $ | 3.1.439.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.439.5t2.a.a | $2$ | $ 439 $ | 5.1.192721.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.439.5t2.a.b | $2$ | $ 439 $ | 5.1.192721.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.3951.10t3.a.b | $2$ | $ 3^{2} \cdot 439 $ | 10.2.3962131404006357.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
| * | 2.3951.10t3.a.a | $2$ | $ 3^{2} \cdot 439 $ | 10.2.3962131404006357.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
| * | 2.439.15t2.a.a | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.3951.30t14.a.c | $2$ | $ 3^{2} \cdot 439 $ | 30.0.141684422447224323645012862864040611366932987.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
| * | 2.439.15t2.a.c | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.3951.30t14.a.a | $2$ | $ 3^{2} \cdot 439 $ | 30.0.141684422447224323645012862864040611366932987.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
| * | 2.439.15t2.a.b | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.3951.30t14.a.b | $2$ | $ 3^{2} \cdot 439 $ | 30.0.141684422447224323645012862864040611366932987.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
| * | 2.439.15t2.a.d | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.3951.30t14.a.d | $2$ | $ 3^{2} \cdot 439 $ | 30.0.141684422447224323645012862864040611366932987.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.