Properties

Label 38.0.566...896.1
Degree $38$
Signature $[0, 19]$
Discriminant $-5.663\times 10^{98}$
Root discriminant $396.98$
Ramified primes $2, 229$
Class number not computed
Class group not computed
Galois group $C_{38}$ (as 38T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^38 + 229*x^36 + 22442*x^34 + 1242325*x^32 + 43174744*x^30 + 989928070*x^28 + 15310092242*x^26 + 160462320009*x^24 + 1129846290713*x^22 + 5241767939799*x^20 + 15619544332617*x^18 + 29338707057674*x^16 + 34962128493052*x^14 + 26729901012072*x^12 + 13161379227544*x^10 + 4139430890096*x^8 + 811133080337*x^6 + 94183964323*x^4 + 5864318791*x^2 + 149876149)
 
gp: K = bnfinit(x^38 + 229*x^36 + 22442*x^34 + 1242325*x^32 + 43174744*x^30 + 989928070*x^28 + 15310092242*x^26 + 160462320009*x^24 + 1129846290713*x^22 + 5241767939799*x^20 + 15619544332617*x^18 + 29338707057674*x^16 + 34962128493052*x^14 + 26729901012072*x^12 + 13161379227544*x^10 + 4139430890096*x^8 + 811133080337*x^6 + 94183964323*x^4 + 5864318791*x^2 + 149876149, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![149876149, 0, 5864318791, 0, 94183964323, 0, 811133080337, 0, 4139430890096, 0, 13161379227544, 0, 26729901012072, 0, 34962128493052, 0, 29338707057674, 0, 15619544332617, 0, 5241767939799, 0, 1129846290713, 0, 160462320009, 0, 15310092242, 0, 989928070, 0, 43174744, 0, 1242325, 0, 22442, 0, 229, 0, 1]);
 

\( x^{38} + 229 x^{36} + 22442 x^{34} + 1242325 x^{32} + 43174744 x^{30} + 989928070 x^{28} + 15310092242 x^{26} + 160462320009 x^{24} + 1129846290713 x^{22} + 5241767939799 x^{20} + 15619544332617 x^{18} + 29338707057674 x^{16} + 34962128493052 x^{14} + 26729901012072 x^{12} + 13161379227544 x^{10} + 4139430890096 x^{8} + 811133080337 x^{6} + 94183964323 x^{4} + 5864318791 x^{2} + 149876149 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $38$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 19]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-56\!\cdots\!896\)\(\medspace = -\,2^{38}\cdot 229^{37}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $396.98$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 229$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $38$
This field is Galois and abelian over $\Q$.
Conductor:  \(916=2^{2}\cdot 229\)
Dirichlet character group:    $\lbrace$$\chi_{916}(1,·)$, $\chi_{916}(643,·)$, $\chi_{916}(901,·)$, $\chi_{916}(905,·)$, $\chi_{916}(11,·)$, $\chi_{916}(15,·)$, $\chi_{916}(17,·)$, $\chi_{916}(899,·)$, $\chi_{916}(661,·)$, $\chi_{916}(795,·)$, $\chi_{916}(671,·)$, $\chi_{916}(161,·)$, $\chi_{916}(165,·)$, $\chi_{916}(431,·)$, $\chi_{916}(691,·)$, $\chi_{916}(53,·)$, $\chi_{916}(57,·)$, $\chi_{916}(415,·)$, $\chi_{916}(61,·)$, $\chi_{916}(501,·)$, $\chi_{916}(583,·)$, $\chi_{916}(333,·)$, $\chi_{916}(855,·)$, $\chi_{916}(729,·)$, $\chi_{916}(859,·)$, $\chi_{916}(915,·)$, $\chi_{916}(863,·)$, $\chi_{916}(627,·)$, $\chi_{916}(225,·)$, $\chi_{916}(187,·)$, $\chi_{916}(485,·)$, $\chi_{916}(273,·)$, $\chi_{916}(289,·)$, $\chi_{916}(751,·)$, $\chi_{916}(755,·)$, $\chi_{916}(245,·)$, $\chi_{916}(121,·)$, $\chi_{916}(255,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{107} a^{26} - \frac{6}{107} a^{24} + \frac{53}{107} a^{22} + \frac{14}{107} a^{20} - \frac{34}{107} a^{18} + \frac{41}{107} a^{16} + \frac{30}{107} a^{14} + \frac{23}{107} a^{12} + \frac{16}{107} a^{8} - \frac{46}{107} a^{6} - \frac{2}{107} a^{4} + \frac{29}{107} a^{2} - \frac{19}{107}$, $\frac{1}{107} a^{27} - \frac{6}{107} a^{25} + \frac{53}{107} a^{23} + \frac{14}{107} a^{21} - \frac{34}{107} a^{19} + \frac{41}{107} a^{17} + \frac{30}{107} a^{15} + \frac{23}{107} a^{13} + \frac{16}{107} a^{9} - \frac{46}{107} a^{7} - \frac{2}{107} a^{5} + \frac{29}{107} a^{3} - \frac{19}{107} a$, $\frac{1}{107} a^{28} + \frac{17}{107} a^{24} + \frac{11}{107} a^{22} + \frac{50}{107} a^{20} + \frac{51}{107} a^{18} - \frac{45}{107} a^{16} - \frac{11}{107} a^{14} + \frac{31}{107} a^{12} + \frac{16}{107} a^{10} + \frac{50}{107} a^{8} + \frac{43}{107} a^{6} + \frac{17}{107} a^{4} + \frac{48}{107} a^{2} - \frac{7}{107}$, $\frac{1}{107} a^{29} + \frac{17}{107} a^{25} + \frac{11}{107} a^{23} + \frac{50}{107} a^{21} + \frac{51}{107} a^{19} - \frac{45}{107} a^{17} - \frac{11}{107} a^{15} + \frac{31}{107} a^{13} + \frac{16}{107} a^{11} + \frac{50}{107} a^{9} + \frac{43}{107} a^{7} + \frac{17}{107} a^{5} + \frac{48}{107} a^{3} - \frac{7}{107} a$, $\frac{1}{107} a^{30} + \frac{6}{107} a^{24} + \frac{5}{107} a^{22} + \frac{27}{107} a^{20} - \frac{2}{107} a^{18} + \frac{41}{107} a^{16} - \frac{51}{107} a^{14} + \frac{53}{107} a^{12} + \frac{50}{107} a^{10} - \frac{15}{107} a^{8} + \frac{50}{107} a^{6} - \frac{25}{107} a^{4} + \frac{35}{107} a^{2} + \frac{2}{107}$, $\frac{1}{107} a^{31} + \frac{6}{107} a^{25} + \frac{5}{107} a^{23} + \frac{27}{107} a^{21} - \frac{2}{107} a^{19} + \frac{41}{107} a^{17} - \frac{51}{107} a^{15} + \frac{53}{107} a^{13} + \frac{50}{107} a^{11} - \frac{15}{107} a^{9} + \frac{50}{107} a^{7} - \frac{25}{107} a^{5} + \frac{35}{107} a^{3} + \frac{2}{107} a$, $\frac{1}{9523} a^{32} + \frac{3}{9523} a^{30} - \frac{6}{9523} a^{28} + \frac{8}{9523} a^{26} - \frac{3408}{9523} a^{24} - \frac{2165}{9523} a^{22} + \frac{2268}{9523} a^{20} - \frac{4084}{9523} a^{18} - \frac{860}{9523} a^{16} - \frac{3933}{9523} a^{14} - \frac{2499}{9523} a^{12} + \frac{4319}{9523} a^{10} - \frac{3045}{9523} a^{8} + \frac{3092}{9523} a^{6} + \frac{389}{9523} a^{4} + \frac{2766}{9523} a^{2} - \frac{4163}{9523}$, $\frac{1}{9523} a^{33} + \frac{3}{9523} a^{31} - \frac{6}{9523} a^{29} + \frac{8}{9523} a^{27} - \frac{3408}{9523} a^{25} - \frac{2165}{9523} a^{23} + \frac{2268}{9523} a^{21} - \frac{4084}{9523} a^{19} - \frac{860}{9523} a^{17} - \frac{3933}{9523} a^{15} - \frac{2499}{9523} a^{13} + \frac{4319}{9523} a^{11} - \frac{3045}{9523} a^{9} + \frac{3092}{9523} a^{7} + \frac{389}{9523} a^{5} + \frac{2766}{9523} a^{3} - \frac{4163}{9523} a$, $\frac{1}{127495457203} a^{34} - \frac{3762954}{127495457203} a^{32} + \frac{594272018}{127495457203} a^{30} - \frac{271586296}{127495457203} a^{28} + \frac{458836508}{127495457203} a^{26} - \frac{37659860813}{127495457203} a^{24} - \frac{21551733898}{127495457203} a^{22} - \frac{54484420657}{127495457203} a^{20} + \frac{21826424541}{127495457203} a^{18} + \frac{61232976004}{127495457203} a^{16} + \frac{9113298200}{127495457203} a^{14} - \frac{39977086622}{127495457203} a^{12} + \frac{22236079087}{127495457203} a^{10} + \frac{15642261184}{127495457203} a^{8} + \frac{34855167983}{127495457203} a^{6} - \frac{37594057250}{127495457203} a^{4} + \frac{20944557106}{127495457203} a^{2} - \frac{35681719383}{127495457203}$, $\frac{1}{127495457203} a^{35} - \frac{3762954}{127495457203} a^{33} + \frac{594272018}{127495457203} a^{31} - \frac{271586296}{127495457203} a^{29} + \frac{458836508}{127495457203} a^{27} - \frac{37659860813}{127495457203} a^{25} - \frac{21551733898}{127495457203} a^{23} - \frac{54484420657}{127495457203} a^{21} + \frac{21826424541}{127495457203} a^{19} + \frac{61232976004}{127495457203} a^{17} + \frac{9113298200}{127495457203} a^{15} - \frac{39977086622}{127495457203} a^{13} + \frac{22236079087}{127495457203} a^{11} + \frac{15642261184}{127495457203} a^{9} + \frac{34855167983}{127495457203} a^{7} - \frac{37594057250}{127495457203} a^{5} + \frac{20944557106}{127495457203} a^{3} - \frac{35681719383}{127495457203} a$, $\frac{1}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{36} + \frac{4611253692845769111308356238387866473384938104800730335938611091412159257939755154580385}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{34} - \frac{17464721208697311416708119177750349072413250427755502818173537446957003487441325683694136639596}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{32} - \frac{9661832890699945846938743870715244443965429536382423719560421552557142255240069591151484378702747}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{30} - \frac{13031126787119119783525053205342901613421455845493608178112847514817351605229077046730954491114117}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{28} + \frac{8846649801369273351511600161653624616690014832787149562784616170739449310563329020749337721667570}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{26} - \frac{764550873005830203861360859231676302870463204474909893917017458852146058202474126405873958611310774}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{24} - \frac{1430710338483021511672661881900638327759115630113851391651146390656323575292549051616052908019979518}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{22} + \frac{1926214617560046204315498347151784355039323368028639264411214045315898983666018747077176970397297901}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{20} + \frac{647827209920609089657432230233261121948526888897130409678998159165847972992795864402883682335873247}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{18} + \frac{1031335037142080859690142341524248129925363955818574823626874036387039821869952786917406153297733522}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{16} + \frac{1592000171084196763973161066291828484124568753336168845430393583619286143523392188225274936050945872}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{14} + \frac{1286465436741378669735362307357378844256401817531791865182348894356000930715212868923570749260931996}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{12} - \frac{905541526634273727165817874051357912990216142482110048673831320500380597181281168706507083389848713}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{10} + \frac{1846112821380756640488786826132873220705914922391367349714819183617399157369181596633658353546580874}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{8} + \frac{1390713646126118611192600100374314797905966134364654811616661137112315131844695499956695683338638962}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{6} - \frac{7611727918084709786761937480961784595060293156217934562560183294598963905803740276080180189354297}{44862863781614138049679089802379192586750927301875205966459061529916871182581668201713213131956777} a^{4} - \frac{1128148921336826268801541015912812308649144450816119640079249783218227932739573299526716008896308454}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153} a^{2} - \frac{74504599672988349086887289655856693419572440095374750241261647919281748665159910608433310040350427}{3992794876563658286421438992411748140220832529866893331014856476162601535249768469952475968744153153}$, $\frac{1}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{37} + \frac{11967764108769065746365219762028664567775234901628424241504494217954306091802093757562545067}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{35} + \frac{8633466287802686498416557265099568805667440312024875645044101690190183320259089519753182735953584}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{33} + \frac{14649244644271964504805803930329711260823023287546803025253157815098800162054604200158570244167228395}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{31} - \frac{14032501039572589624639983204757354666057124971279789914545927015896710996811544417758294686119224957}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{29} + \frac{889681334011398371049649047179295860792243719354488434341993153635789123310752421271526382738654325}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{27} + \frac{1582552644561238807453898198183423222753192941039999371557919811606430118874923880099743311498683508591}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{25} - \frac{704461782652396682760795124327100948301851586216591838254953266186521700090096380785639918218350846517}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{23} + \frac{619599767877622067821517062611636444738233533909925045516053964619530905287160781670779367951093640053}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{21} - \frac{804935665558834434950392081252026701684686240635810235033167123365885374314975143448951120395830547111}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{19} + \frac{1489557839281893527835722814757487929740415780806230723810398661009622407150132422892955453805977749154}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{17} + \frac{1407063236391761297254065345251390018013216679496356166023039538477813538377314771763292613021856470829}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{15} + \frac{778150229122328577973988712193039848259862819456051251027553168557668921267863521015534789975452673591}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{13} - \frac{674370609590381177534666881468181778336320419497076528246982800779157858986789776750522674525039251079}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{11} - \frac{464696853446759364577285566866570608398623391411493342233197588346409877821541322731506957913361518700}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{9} + \frac{528559156487571872718217201482370495601272418450295525163040326251864644216017123165164628958585054278}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{7} + \frac{635636499759500420504235498386929892134219977384670121493325802174148767368909570666116462002231923934}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{5} - \frac{222720018219425982012422332295352454122230247971043065975179976668005577499570291110003499462758964183}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a^{3} - \frac{1069639884338431399868189943058743889449360705125452816443679845386559570284453167861326759556370217581}{3230171055139999553714944144861104245438653516662316704791018889215544642017062692191553058714019900777} a$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $18$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_{38}$ (as 38T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 38
The 38 conjugacy class representatives for $C_{38}$
Character table for $C_{38}$ is not computed

Intermediate fields

\(\Q(\sqrt{-229}) \), 19.19.2999429662895796650415561622892044448017561.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $38$ $19^{2}$ $19^{2}$ $38$ $38$ $19^{2}$ $38$ $19^{2}$ $38$ $19^{2}$ $19^{2}$ $38$ $38$ $19^{2}$ $19^{2}$ $19^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
229Data not computed