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Properties

Label 32.0.15729269092...000.14
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $61.29$
Ramified primes $2, 3, 5, 7$
Class number $640$ (GRH)
Class group $[2, 2, 2, 4, 20]$ (GRH)
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78961, -594596, 2169322, -5760912, 13175136, -25509340, 41428692, -60196108, 79406215, -90665588, 92328106, -89459536, 76409356, -54449344, 39573806, -29635068, 15048996, -5005052, 4137624, -3519844, 915936, 123100, 129290, -160544, 22235, 10276, 1252, -2284, 236, 80, 4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 4*x^30 + 80*x^29 + 236*x^28 - 2284*x^27 + 1252*x^26 + 10276*x^25 + 22235*x^24 - 160544*x^23 + 129290*x^22 + 123100*x^21 + 915936*x^20 - 3519844*x^19 + 4137624*x^18 - 5005052*x^17 + 15048996*x^16 - 29635068*x^15 + 39573806*x^14 - 54449344*x^13 + 76409356*x^12 - 89459536*x^11 + 92328106*x^10 - 90665588*x^9 + 79406215*x^8 - 60196108*x^7 + 41428692*x^6 - 25509340*x^5 + 13175136*x^4 - 5760912*x^3 + 2169322*x^2 - 594596*x + 78961)
 
gp: K = bnfinit(x^32 - 8*x^31 + 4*x^30 + 80*x^29 + 236*x^28 - 2284*x^27 + 1252*x^26 + 10276*x^25 + 22235*x^24 - 160544*x^23 + 129290*x^22 + 123100*x^21 + 915936*x^20 - 3519844*x^19 + 4137624*x^18 - 5005052*x^17 + 15048996*x^16 - 29635068*x^15 + 39573806*x^14 - 54449344*x^13 + 76409356*x^12 - 89459536*x^11 + 92328106*x^10 - 90665588*x^9 + 79406215*x^8 - 60196108*x^7 + 41428692*x^6 - 25509340*x^5 + 13175136*x^4 - 5760912*x^3 + 2169322*x^2 - 594596*x + 78961, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} + 4 x^{30} + 80 x^{29} + 236 x^{28} - 2284 x^{27} + 1252 x^{26} + 10276 x^{25} + 22235 x^{24} - 160544 x^{23} + 129290 x^{22} + 123100 x^{21} + 915936 x^{20} - 3519844 x^{19} + 4137624 x^{18} - 5005052 x^{17} + 15048996 x^{16} - 29635068 x^{15} + 39573806 x^{14} - 54449344 x^{13} + 76409356 x^{12} - 89459536 x^{11} + 92328106 x^{10} - 90665588 x^{9} + 79406215 x^{8} - 60196108 x^{7} + 41428692 x^{6} - 25509340 x^{5} + 13175136 x^{4} - 5760912 x^{3} + 2169322 x^{2} - 594596 x + 78961 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.29$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(517,·)$, $\chi_{840}(769,·)$, $\chi_{840}(139,·)$, $\chi_{840}(13,·)$, $\chi_{840}(659,·)$, $\chi_{840}(533,·)$, $\chi_{840}(407,·)$, $\chi_{840}(281,·)$, $\chi_{840}(797,·)$, $\chi_{840}(419,·)$, $\chi_{840}(293,·)$, $\chi_{840}(167,·)$, $\chi_{840}(41,·)$, $\chi_{840}(811,·)$, $\chi_{840}(169,·)$, $\chi_{840}(449,·)$, $\chi_{840}(197,·)$, $\chi_{840}(463,·)$, $\chi_{840}(209,·)$, $\chi_{840}(211,·)$, $\chi_{840}(727,·)$, $\chi_{840}(601,·)$, $\chi_{840}(223,·)$, $\chi_{840}(379,·)$, $\chi_{840}(743,·)$, $\chi_{840}(491,·)$, $\chi_{840}(757,·)$, $\chi_{840}(503,·)$, $\chi_{840}(251,·)$, $\chi_{840}(253,·)$, $\chi_{840}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{11}$, $\frac{1}{4} a^{24} - \frac{1}{4}$, $\frac{1}{4} a^{25} - \frac{1}{4} a$, $\frac{1}{44} a^{26} - \frac{1}{11} a^{25} - \frac{5}{44} a^{24} - \frac{1}{11} a^{23} + \frac{2}{11} a^{22} + \frac{5}{22} a^{21} + \frac{3}{22} a^{20} + \frac{1}{11} a^{19} - \frac{2}{11} a^{18} + \frac{3}{22} a^{17} + \frac{1}{11} a^{16} - \frac{1}{22} a^{15} - \frac{5}{22} a^{14} - \frac{1}{22} a^{13} - \frac{1}{11} a^{11} + \frac{2}{11} a^{10} - \frac{1}{2} a^{9} - \frac{5}{22} a^{8} + \frac{5}{11} a^{7} - \frac{5}{11} a^{6} - \frac{9}{22} a^{5} - \frac{5}{11} a^{4} - \frac{1}{2} a^{3} + \frac{9}{44} a^{2} - \frac{1}{2} a + \frac{13}{44}$, $\frac{1}{44} a^{27} + \frac{1}{44} a^{25} - \frac{1}{22} a^{24} - \frac{2}{11} a^{23} - \frac{1}{22} a^{22} + \frac{1}{22} a^{21} + \frac{3}{22} a^{20} + \frac{2}{11} a^{19} - \frac{1}{11} a^{18} + \frac{3}{22} a^{17} - \frac{2}{11} a^{16} + \frac{1}{11} a^{15} + \frac{1}{22} a^{14} - \frac{2}{11} a^{13} - \frac{1}{11} a^{12} - \frac{2}{11} a^{11} + \frac{5}{22} a^{10} - \frac{5}{22} a^{9} + \frac{1}{22} a^{8} + \frac{4}{11} a^{7} + \frac{3}{11} a^{6} + \frac{9}{22} a^{5} + \frac{2}{11} a^{4} - \frac{13}{44} a^{3} + \frac{7}{22} a^{2} - \frac{9}{44} a - \frac{7}{22}$, $\frac{1}{9716080} a^{28} + \frac{8279}{2429020} a^{27} + \frac{4043}{2429020} a^{26} + \frac{26519}{220820} a^{25} + \frac{387791}{9716080} a^{24} + \frac{429509}{2429020} a^{23} + \frac{19379}{1214510} a^{22} + \frac{421879}{2429020} a^{21} - \frac{26363}{242902} a^{20} - \frac{3397}{39820} a^{19} + \frac{51649}{971608} a^{18} - \frac{66521}{1214510} a^{17} + \frac{44982}{607255} a^{16} - \frac{4857}{242902} a^{15} + \frac{62079}{971608} a^{14} + \frac{437671}{2429020} a^{13} + \frac{54125}{485804} a^{12} + \frac{585437}{2429020} a^{11} + \frac{18523}{1214510} a^{10} - \frac{20585}{485804} a^{9} - \frac{1766}{9955} a^{8} - \frac{740213}{2429020} a^{7} - \frac{465447}{971608} a^{6} - \frac{65271}{242902} a^{5} - \frac{3848577}{9716080} a^{4} + \frac{125879}{2429020} a^{3} + \frac{1923909}{4858040} a^{2} - \frac{65519}{1214510} a - \frac{4503779}{9716080}$, $\frac{1}{9716080} a^{29} - \frac{15291}{2429020} a^{27} - \frac{179}{2429020} a^{26} - \frac{205589}{1943216} a^{25} - \frac{17409}{485804} a^{24} - \frac{58464}{607255} a^{23} + \frac{96971}{2429020} a^{22} - \frac{74007}{1214510} a^{21} - \frac{506887}{2429020} a^{20} + \frac{249949}{4858040} a^{19} + \frac{61592}{607255} a^{18} + \frac{9908}{121451} a^{17} + \frac{31331}{1214510} a^{16} + \frac{191067}{971608} a^{15} + \frac{61121}{2429020} a^{14} + \frac{540949}{2429020} a^{13} + \frac{547637}{2429020} a^{12} + \frac{37326}{607255} a^{11} - \frac{1146441}{2429020} a^{10} + \frac{22764}{55205} a^{9} - \frac{863799}{2429020} a^{8} + \frac{95261}{4858040} a^{7} + \frac{30721}{121451} a^{6} - \frac{124307}{883280} a^{5} + \frac{409351}{1214510} a^{4} + \frac{2427541}{4858040} a^{3} + \frac{30093}{121451} a^{2} + \frac{2603173}{9716080} a + \frac{250633}{1214510}$, $\frac{1}{7573188839920} a^{30} - \frac{239607}{7573188839920} a^{29} - \frac{27}{1261147184} a^{28} + \frac{4103722736}{473324302495} a^{27} - \frac{68739645}{1514637767984} a^{26} + \frac{716161203919}{7573188839920} a^{25} - \frac{14742824081}{1514637767984} a^{24} - \frac{28173072484}{473324302495} a^{23} + \frac{25905811181}{1893297209980} a^{22} + \frac{12371694691}{946648604990} a^{21} - \frac{118123794773}{3786594419960} a^{20} + \frac{229651013747}{3786594419960} a^{19} + \frac{42389046263}{344235856360} a^{18} + \frac{1109261667}{5230102790} a^{17} - \frac{581508471649}{3786594419960} a^{16} + \frac{448566673497}{3786594419960} a^{15} + \frac{490776459159}{3786594419960} a^{14} - \frac{469018085287}{1893297209980} a^{13} + \frac{29137376819}{189329720998} a^{12} + \frac{148578538586}{473324302495} a^{11} + \frac{510588809767}{1893297209980} a^{10} - \frac{45871092884}{473324302495} a^{9} - \frac{201370994333}{757318883992} a^{8} - \frac{161670762141}{3786594419960} a^{7} + \frac{972719852393}{7573188839920} a^{6} - \frac{457511864753}{7573188839920} a^{5} + \frac{2417385648773}{7573188839920} a^{4} - \frac{343861202827}{757318883992} a^{3} - \frac{753949541093}{1514637767984} a^{2} - \frac{701325139799}{1514637767984} a - \frac{2905226734459}{7573188839920}$, $\frac{1}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{31} - \frac{103044410706598622544615493137157343296761563977427235153887819067}{3517315886601928072509240632247994652385058832981226768723775476919344393366320} a^{30} + \frac{1961282274326595275216401063834707392282080935633024203341036302468861851}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{29} - \frac{1414674220150481264068909279384248232425792689192295703655060930334847937}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{28} + \frac{396192131178799103017791108257107974166640139584937009538698236273046480688203}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{27} - \frac{226949967239056975870716284575772910902613880496529048037207086580317433686083}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{26} + \frac{59280717130760344334219558113200568944570584467727565387938404526708496512027}{3517315886601928072509240632247994652385058832981226768723775476919344393366320} a^{25} + \frac{251082690660504729984426634440665703428995397369394931560938012566564971607793}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{24} - \frac{5875866742427384047425641874759107411075734283981941939196859775923172256728}{219832242912620504531827539515499665774066177061326673045235967307459024585395} a^{23} + \frac{384330440803210394157859948706965889180933046325961410249275255309527264670376}{2418154672038825549850102934670496323514727947674593403497595640382049270439345} a^{22} + \frac{104890251862570406991309172120855388126812226865665047562148797752924647124721}{3869047475262120879760164695472794117623564716279349445596153024611278832702952} a^{21} + \frac{687748621481178456042432074497450779720836632591228563827219244395921343571453}{19345237376310604398800823477363970588117823581396747227980765123056394163514760} a^{20} + \frac{4459351722029884320747580187122994598478946061528569198263176946387501322282029}{19345237376310604398800823477363970588117823581396747227980765123056394163514760} a^{19} + \frac{2069456444112783120329305146190343981885451290392067834720551538653083242363899}{19345237376310604398800823477363970588117823581396747227980765123056394163514760} a^{18} - \frac{712915153810070494286321529984805671897211103642810657322586678010482235461397}{19345237376310604398800823477363970588117823581396747227980765123056394163514760} a^{17} + \frac{51857090203724949109269906769101300867654322373825142387576380448637727462913}{1758657943300964036254620316123997326192529416490613384361887738459672196683160} a^{16} + \frac{3098830745378924696095208220927454782973009510761363404147127833001590085930069}{19345237376310604398800823477363970588117823581396747227980765123056394163514760} a^{15} + \frac{3558023911382819222466623995361838924099270704357335203267281281146678929623583}{19345237376310604398800823477363970588117823581396747227980765123056394163514760} a^{14} + \frac{456270656434954450683482571148432583302235449556769851071503865773181938095801}{9672618688155302199400411738681985294058911790698373613990382561528197081757380} a^{13} + \frac{22751577182427520587396812430675874213796193310699109970702669438889333364391}{9672618688155302199400411738681985294058911790698373613990382561528197081757380} a^{12} + \frac{16367283960251293502314031818691646276402820972125403722982084276360073795585}{33353857545363111032415212892006845841582454450684046944794422625959300281922} a^{11} + \frac{30240832815896934407096950084992095491061564175838706031009203412037580854149}{87932897165048201812731015806199866309626470824530669218094386922983609834158} a^{10} + \frac{603904882317450813430343380774039734533675004390837109715559103816160302419707}{1758657943300964036254620316123997326192529416490613384361887738459672196683160} a^{9} - \frac{1558105235286313651662300189302884769424054542361768398353799997612777086082099}{3869047475262120879760164695472794117623564716279349445596153024611278832702952} a^{8} - \frac{10482379798956011656529877067804234540778977885876845408271475834620617484061879}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{7} + \frac{2470285508677538150973893303785989408622124569156136857808797910867917522590287}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{6} - \frac{5289666861746460382821173373363569847061035056924895212115632474379044523666869}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{5} - \frac{8581287864715196262872809031046051496607379206953304476987700957947560789488441}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{4} - \frac{14323988133520482003380952585583119159988275475317995883141397056240024514269961}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{3} + \frac{14710832505989426243385103941576436527525294118643241186994414890187154672199961}{38690474752621208797601646954727941176235647162793494455961530246112788327029520} a^{2} - \frac{2244444678247690861527680917045378671923750870775161899165624646504741909688153}{7738094950524241759520329390945588235247129432558698891192306049222557665405904} a - \frac{67483925383243943907555582657708119128627287034784027831850245714300581548709}{137688522251321027749472053219672388527528993461898556782781246427447645291920}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{20}$, which has order $640$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1967536555511405360629789013828985637985048527831667536444286209792339}{1458890330598863504728219002758997410991084067129454677933890124851908360} a^{31} + \frac{14299948417852391607788785944098161634547211809551759065537298544125847}{1458890330598863504728219002758997410991084067129454677933890124851908360} a^{30} + \frac{2583383641572907981969260577067980110200697182472794599289011837990387}{1458890330598863504728219002758997410991084067129454677933890124851908360} a^{29} - \frac{310811071594306861868385428928142488732301612735577217134094217638073949}{2917780661197727009456438005517994821982168134258909355867780249703816720} a^{28} - \frac{578066738316786506589122814874237562076347384253130392386875784004825329}{1458890330598863504728219002758997410991084067129454677933890124851908360} a^{27} + \frac{4069470747738026368391958999427619025402155667897228578391551100387734213}{1458890330598863504728219002758997410991084067129454677933890124851908360} a^{26} + \frac{511158531127607814071712122221303567802244099138390600211374514323409913}{1458890330598863504728219002758997410991084067129454677933890124851908360} a^{25} - \frac{39627136187095611769934606490877035051349972687486590071521321859353370999}{2917780661197727009456438005517994821982168134258909355867780249703816720} a^{24} - \frac{5824285803094017493468073657503517739714882109996805464360906256282652941}{145889033059886350472821900275899741099108406712945467793389012485190836} a^{23} + \frac{34135139122771354823836693969127744197904500710324775936713794764905307506}{182361291324857938091027375344874676373885508391181834741736265606488545} a^{22} - \frac{13737370350190458508748839871487376019574916863203965528437975583968177883}{364722582649715876182054750689749352747771016782363669483472531212977090} a^{21} - \frac{28038053664768439281876320733428870019888623038760412644736506178961575911}{145889033059886350472821900275899741099108406712945467793389012485190836} a^{20} - \frac{501823052740662884925815940184185329349358646237443887677899497662126848351}{364722582649715876182054750689749352747771016782363669483472531212977090} a^{19} + \frac{1090679194820954675064500969872672443026018645338046444429238312904717820651}{291778066119772700945643800551799482198216813425890935586778024970381672} a^{18} - \frac{416505058269576417392135394360561411606690382124341936092343596206054709927}{145889033059886350472821900275899741099108406712945467793389012485190836} a^{17} + \frac{3418867082951380434611009343574125529235601490179778035468777847212612728413}{729445165299431752364109501379498705495542033564727338966945062425954180} a^{16} - \frac{12311817306056059589114062425426415951408491480402349475073738352965038759313}{729445165299431752364109501379498705495542033564727338966945062425954180} a^{15} + \frac{40325993743845142872572096023036370326480379492967636242901445052276924439129}{1458890330598863504728219002758997410991084067129454677933890124851908360} a^{14} - \frac{24262372100669582275174005567956585590784710705387017947594887363663658132387}{729445165299431752364109501379498705495542033564727338966945062425954180} a^{13} + \frac{35935162798050449321046842951554039695262503050197722859857804687456103562143}{729445165299431752364109501379498705495542033564727338966945062425954180} a^{12} - \frac{1689923437798330993788063036029932414265325062212978320939947222948030931269}{25153281562049370771176189702741334672260070122921632378170519393998420} a^{11} + \frac{13091894411433631360467119625773147040582909644536822198653992345299313598152}{182361291324857938091027375344874676373885508391181834741736265606488545} a^{10} - \frac{26402902724643438176128767469447142905038202690951021590428626503693829407879}{364722582649715876182054750689749352747771016782363669483472531212977090} a^{9} + \frac{50845527637602577447661744883859508789154405283679926320675186564271240901203}{729445165299431752364109501379498705495542033564727338966945062425954180} a^{8} - \frac{82421633906760030152589990999163693433899035722277355421592593028954718644409}{1458890330598863504728219002758997410991084067129454677933890124851908360} a^{7} + \frac{667042554655376085333310155153732232756650167241475059079996362473941909923}{16578299211350721644638852304079516033989591671925621340157842327862595} a^{6} - \frac{660883031331473784635305487716294477864727837519814917586434869759690466011}{24726954755912940758105406826423684932052272324228045388710002116134040} a^{5} + \frac{43920577569483169852231379339589091715716073939644019352761641204861212727361}{2917780661197727009456438005517994821982168134258909355867780249703816720} a^{4} - \frac{10052369423392419117494793702533206325312835088537987593459958202257477015313}{1458890330598863504728219002758997410991084067129454677933890124851908360} a^{3} + \frac{1020363952627023173240902098916592545691234014326303823897975024289148460149}{364722582649715876182054750689749352747771016782363669483472531212977090} a^{2} - \frac{1326261367464600964925559611479081136817883699274875755463874409732252266943}{1458890330598863504728219002758997410991084067129454677933890124851908360} a + \frac{1556727447794322334611695926825766892250623348661206878048072059614773507}{10383561071878032062122555179779341003495260264266581337607758895743120} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28949477018021.215 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3\times C_4$ (as 32T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-210}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(\sqrt{6}, \sqrt{-35})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{105})\), \(\Q(\sqrt{6}, \sqrt{70})\), \(\Q(\sqrt{-3}, \sqrt{70})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-10}, \sqrt{14})\), \(\Q(\sqrt{30}, \sqrt{-35})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{-7}, \sqrt{-10})\), \(\Q(\sqrt{21}, \sqrt{30})\), \(\Q(\sqrt{-15}, \sqrt{-42})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{-42})\), \(\Q(\sqrt{-7}, \sqrt{30})\), \(\Q(\sqrt{-10}, \sqrt{21})\), \(\Q(\sqrt{14}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{-10}, \sqrt{-42})\), \(\Q(\sqrt{14}, \sqrt{30})\), 4.0.8000.2, 4.4.392000.1, \(\Q(\zeta_{20})^+\), 4.0.98000.1, 4.0.18000.1, 4.4.882000.1, 4.4.72000.1, 4.0.3528000.1, 8.0.497871360000.17, 8.0.6146560000.1, 8.0.497871360000.8, 8.0.497871360000.15, 8.0.497871360000.4, 8.0.121550625.1, 8.0.497871360000.12, 8.0.207360000.2, 8.0.796594176.1, 8.0.497871360000.13, 8.0.497871360000.3, 8.8.497871360000.2, 8.0.497871360000.2, 8.0.497871360000.20, 8.0.497871360000.18, 8.0.153664000000.2, 8.0.9604000000.2, 8.0.777924000000.4, 8.0.12446784000000.15, 8.0.1024000000.1, 8.0.2458624000000.5, 8.0.82944000000.5, 8.0.199148544000000.169, 8.0.2458624000000.1, 8.8.2458624000000.2, 8.0.199148544000000.163, 8.8.199148544000000.3, 8.0.82944000000.4, 8.8.199148544000000.10, 8.8.82944000000.2, 8.0.199148544000000.168, 8.0.199148544000000.59, 8.0.199148544000000.219, 8.0.199148544000000.160, 8.0.199148544000000.142, 8.0.5184000000.3, 8.0.12446784000000.18, 8.0.324000000.2, 8.0.777924000000.9, 8.0.12446784000000.5, 8.8.12446784000000.3, 8.8.777924000000.2, 8.0.777924000000.1, 16.0.247875891108249600000000.2, 16.0.6044831973376000000000000.9, 16.0.39660142577319936000000000000.18, 16.0.39660142577319936000000000000.65, 16.0.39660142577319936000000000000.34, 16.0.154922431942656000000000000.14, 16.0.605165749776000000000000.10, 16.0.6879707136000000000000.8, 16.0.39660142577319936000000000000.53, 16.0.39660142577319936000000000000.42, 16.0.39660142577319936000000000000.33, 16.0.39660142577319936000000000000.6, 16.16.39660142577319936000000000000.7, 16.0.39660142577319936000000000000.24, 16.0.39660142577319936000000000000.47

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 R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.16.1$x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 12$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.1$x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 12$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.1$x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 12$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.1$x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 12$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$