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Properties

Label 32.0.13927098107...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $39.65$
Ramified primes $3, 5, 13$
Class number $104$ (GRH)
Class group $[104]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, 8748, 4374, 6318, 13527, 37152, 32913, 15519, 26164, 32414, 89951, 81350, 25190, 29982, 50069, 118245, 47769, -39009, 1484, -10201, 9166, -1297, 1343, -1272, 228, -237, 134, -18, 20, -13, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + 2*x^30 - 13*x^29 + 20*x^28 - 18*x^27 + 134*x^26 - 237*x^25 + 228*x^24 - 1272*x^23 + 1343*x^22 - 1297*x^21 + 9166*x^20 - 10201*x^19 + 1484*x^18 - 39009*x^17 + 47769*x^16 + 118245*x^15 + 50069*x^14 + 29982*x^13 + 25190*x^12 + 81350*x^11 + 89951*x^10 + 32414*x^9 + 26164*x^8 + 15519*x^7 + 32913*x^6 + 37152*x^5 + 13527*x^4 + 6318*x^3 + 4374*x^2 + 8748*x + 6561)
 
gp: K = bnfinit(x^32 - x^31 + 2*x^30 - 13*x^29 + 20*x^28 - 18*x^27 + 134*x^26 - 237*x^25 + 228*x^24 - 1272*x^23 + 1343*x^22 - 1297*x^21 + 9166*x^20 - 10201*x^19 + 1484*x^18 - 39009*x^17 + 47769*x^16 + 118245*x^15 + 50069*x^14 + 29982*x^13 + 25190*x^12 + 81350*x^11 + 89951*x^10 + 32414*x^9 + 26164*x^8 + 15519*x^7 + 32913*x^6 + 37152*x^5 + 13527*x^4 + 6318*x^3 + 4374*x^2 + 8748*x + 6561, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} + 2 x^{30} - 13 x^{29} + 20 x^{28} - 18 x^{27} + 134 x^{26} - 237 x^{25} + 228 x^{24} - 1272 x^{23} + 1343 x^{22} - 1297 x^{21} + 9166 x^{20} - 10201 x^{19} + 1484 x^{18} - 39009 x^{17} + 47769 x^{16} + 118245 x^{15} + 50069 x^{14} + 29982 x^{13} + 25190 x^{12} + 81350 x^{11} + 89951 x^{10} + 32414 x^{9} + 26164 x^{8} + 15519 x^{7} + 32913 x^{6} + 37152 x^{5} + 13527 x^{4} + 6318 x^{3} + 4374 x^{2} + 8748 x + 6561 \)

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:  \(1392709810786878646248924865118341505527496337890625=3^{16}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.65$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(1,·)$, $\chi_{195}(131,·)$, $\chi_{195}(8,·)$, $\chi_{195}(14,·)$, $\chi_{195}(148,·)$, $\chi_{195}(151,·)$, $\chi_{195}(157,·)$, $\chi_{195}(31,·)$, $\chi_{195}(161,·)$, $\chi_{195}(34,·)$, $\chi_{195}(164,·)$, $\chi_{195}(38,·)$, $\chi_{195}(44,·)$, $\chi_{195}(47,·)$, $\chi_{195}(53,·)$, $\chi_{195}(187,·)$, $\chi_{195}(181,·)$, $\chi_{195}(64,·)$, $\chi_{195}(194,·)$, $\chi_{195}(73,·)$, $\chi_{195}(77,·)$, $\chi_{195}(79,·)$, $\chi_{195}(83,·)$, $\chi_{195}(142,·)$, $\chi_{195}(86,·)$, $\chi_{195}(92,·)$, $\chi_{195}(103,·)$, $\chi_{195}(109,·)$, $\chi_{195}(112,·)$, $\chi_{195}(116,·)$, $\chi_{195}(118,·)$, $\chi_{195}(122,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{18} - \frac{1}{6} a^{16} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{19} - \frac{1}{6} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{20} - \frac{1}{6} a^{16} - \frac{1}{3} a^{14} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{21} - \frac{1}{3} a^{11} - \frac{1}{2} a^{6} - \frac{1}{3} a$, $\frac{1}{18} a^{22} + \frac{1}{18} a^{21} - \frac{1}{18} a^{17} + \frac{2}{9} a^{15} - \frac{1}{9} a^{12} + \frac{1}{3} a^{11} + \frac{2}{9} a^{9} - \frac{7}{18} a^{7} - \frac{1}{2} a^{6} + \frac{2}{9} a^{3} - \frac{5}{18} a^{2} + \frac{1}{3} a$, $\frac{1}{18} a^{23} - \frac{1}{18} a^{21} - \frac{1}{18} a^{18} + \frac{1}{18} a^{17} + \frac{2}{9} a^{16} - \frac{2}{9} a^{15} - \frac{1}{9} a^{13} + \frac{4}{9} a^{12} - \frac{1}{3} a^{11} + \frac{2}{9} a^{10} - \frac{2}{9} a^{9} - \frac{7}{18} a^{8} - \frac{1}{9} a^{7} - \frac{1}{2} a^{6} + \frac{2}{9} a^{4} - \frac{1}{2} a^{3} - \frac{7}{18} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{24} + \frac{1}{18} a^{21} - \frac{1}{18} a^{19} + \frac{1}{18} a^{18} - \frac{2}{9} a^{16} - \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{4}{9} a^{13} - \frac{4}{9} a^{12} - \frac{1}{9} a^{11} - \frac{2}{9} a^{10} - \frac{1}{2} a^{9} - \frac{1}{9} a^{8} + \frac{4}{9} a^{7} - \frac{1}{2} a^{6} + \frac{2}{9} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{1098} a^{25} - \frac{25}{1098} a^{24} + \frac{1}{1098} a^{23} - \frac{7}{366} a^{22} + \frac{14}{183} a^{21} - \frac{37}{1098} a^{20} - \frac{17}{549} a^{19} - \frac{35}{1098} a^{18} + \frac{35}{549} a^{17} + \frac{5}{183} a^{16} + \frac{50}{549} a^{15} + \frac{65}{549} a^{14} + \frac{16}{61} a^{13} - \frac{118}{549} a^{12} - \frac{103}{549} a^{11} - \frac{463}{1098} a^{10} - \frac{181}{1098} a^{9} + \frac{131}{366} a^{8} - \frac{145}{366} a^{7} + \frac{155}{549} a^{6} - \frac{13}{1098} a^{5} - \frac{151}{549} a^{4} - \frac{5}{366} a^{3} + \frac{86}{183} a^{2} - \frac{9}{61} a + \frac{27}{61}$, $\frac{1}{121506876} a^{26} + \frac{1589}{30376719} a^{25} - \frac{343112}{30376719} a^{24} - \frac{338351}{121506876} a^{23} + \frac{1540289}{121506876} a^{22} + \frac{222545}{10125573} a^{21} + \frac{74555}{1991916} a^{20} + \frac{4912751}{60753438} a^{19} - \frac{1391981}{40502292} a^{18} - \frac{2368633}{121506876} a^{17} + \frac{25603703}{121506876} a^{16} + \frac{15052835}{121506876} a^{15} - \frac{6523385}{20251146} a^{14} - \frac{5127922}{30376719} a^{13} + \frac{8052419}{30376719} a^{12} - \frac{46782763}{121506876} a^{11} + \frac{7772683}{20251146} a^{10} - \frac{6115667}{30376719} a^{9} + \frac{36630149}{121506876} a^{8} - \frac{17906891}{40502292} a^{7} - \frac{26300369}{60753438} a^{6} + \frac{33900211}{121506876} a^{5} + \frac{2474977}{30376719} a^{4} - \frac{19252615}{121506876} a^{3} - \frac{55974983}{121506876} a^{2} - \frac{1873983}{13500764} a - \frac{5942635}{13500764}$, $\frac{1}{364520628} a^{27} - \frac{1}{364520628} a^{26} + \frac{1339}{2987874} a^{25} - \frac{6143581}{364520628} a^{24} + \frac{126086}{91130157} a^{23} + \frac{882817}{121506876} a^{22} - \frac{2278549}{364520628} a^{21} - \frac{1692169}{40502292} a^{20} + \frac{2744441}{121506876} a^{19} + \frac{472259}{6750382} a^{18} + \frac{2211842}{91130157} a^{17} + \frac{18545648}{91130157} a^{16} - \frac{20993831}{364520628} a^{15} + \frac{888313}{2987874} a^{14} + \frac{24482561}{91130157} a^{13} + \frac{27303151}{121506876} a^{12} + \frac{52833727}{121506876} a^{11} - \frac{1415787}{3375191} a^{10} + \frac{67139231}{364520628} a^{9} + \frac{4233163}{20251146} a^{8} + \frac{98392463}{364520628} a^{7} - \frac{91195303}{364520628} a^{6} - \frac{128004823}{364520628} a^{5} + \frac{52663253}{364520628} a^{4} + \frac{23913343}{91130157} a^{3} - \frac{8730131}{30376719} a^{2} + \frac{3475208}{10125573} a + \frac{1874515}{13500764}$, $\frac{1}{1093561884} a^{28} - \frac{1}{1093561884} a^{27} + \frac{1}{546780942} a^{26} + \frac{325625}{1093561884} a^{25} + \frac{665564}{273390471} a^{24} + \frac{773647}{40502292} a^{23} - \frac{25458193}{1093561884} a^{22} + \frac{6032009}{364520628} a^{21} - \frac{11686493}{364520628} a^{20} + \frac{4587512}{91130157} a^{19} + \frac{529813}{546780942} a^{18} + \frac{26002453}{546780942} a^{17} - \frac{56042321}{1093561884} a^{16} - \frac{135913979}{546780942} a^{15} + \frac{86677436}{273390471} a^{14} - \frac{15690445}{364520628} a^{13} - \frac{155335681}{364520628} a^{12} - \frac{16309469}{91130157} a^{11} + \frac{418868921}{1093561884} a^{10} + \frac{17592641}{182260314} a^{9} + \frac{171315701}{1093561884} a^{8} + \frac{140364305}{1093561884} a^{7} + \frac{219173045}{1093561884} a^{6} - \frac{6938005}{1093561884} a^{5} - \frac{265066969}{546780942} a^{4} + \frac{28073699}{182260314} a^{3} - \frac{5901233}{20251146} a^{2} - \frac{5733019}{40502292} a + \frac{1222078}{3375191}$, $\frac{1}{49843066541928443748} a^{29} + \frac{16709495321}{49843066541928443748} a^{28} + \frac{15287967269}{49843066541928443748} a^{27} + \frac{710764021}{24921533270964221874} a^{26} - \frac{5272477282869413}{24921533270964221874} a^{25} - \frac{61698588283372474}{4153588878494036979} a^{24} + \frac{402727385411831645}{49843066541928443748} a^{23} - \frac{171218970467218183}{8307177756988073958} a^{22} + \frac{48284907133010128}{4153588878494036979} a^{21} - \frac{1324445519351349439}{16614355513976147916} a^{20} + \frac{4443392956801547}{49843066541928443748} a^{19} + \frac{15045988910707853}{12460766635482110937} a^{18} - \frac{3117087503043679259}{49843066541928443748} a^{17} + \frac{1518816281692646227}{24921533270964221874} a^{16} + \frac{4230200134034748905}{49843066541928443748} a^{15} + \frac{2225792930297747365}{16614355513976147916} a^{14} - \frac{7294894200588482227}{16614355513976147916} a^{13} + \frac{1074142762504957441}{16614355513976147916} a^{12} - \frac{5509399026369171649}{12460766635482110937} a^{11} - \frac{300250009521011219}{2769059252329357986} a^{10} + \frac{2221230203847843923}{12460766635482110937} a^{9} - \frac{1133634574245573895}{49843066541928443748} a^{8} - \frac{2399183280621734107}{12460766635482110937} a^{7} + \frac{1474406854533543767}{12460766635482110937} a^{6} + \frac{11191830307739086255}{49843066541928443748} a^{5} - \frac{1936797795870446701}{16614355513976147916} a^{4} - \frac{1328477479883700385}{2769059252329357986} a^{3} + \frac{859193285906356103}{1846039501552905324} a^{2} + \frac{20899601238783949}{153836625129408777} a + \frac{22144313739471747}{205115500172545036}$, $\frac{1}{149529199625785331244} a^{30} - \frac{1}{149529199625785331244} a^{29} - \frac{38580030619}{149529199625785331244} a^{28} - \frac{47450740963}{149529199625785331244} a^{27} + \frac{8870710349}{149529199625785331244} a^{26} + \frac{711605089398137}{8307177756988073958} a^{25} + \frac{139622890483985701}{74764599812892665622} a^{24} - \frac{263272898191146214}{12460766635482110937} a^{23} - \frac{1171750549630602641}{49843066541928443748} a^{22} - \frac{416754210231750983}{24921533270964221874} a^{21} - \frac{1656183479653548499}{37382299906446332811} a^{20} + \frac{3806618617312754237}{149529199625785331244} a^{19} - \frac{620980488412756133}{149529199625785331244} a^{18} + \frac{377400600863615239}{74764599812892665622} a^{17} + \frac{28166955003910305845}{149529199625785331244} a^{16} + \frac{2945280924545073476}{12460766635482110937} a^{15} + \frac{14933386625689014905}{49843066541928443748} a^{14} + \frac{7203588423313743661}{49843066541928443748} a^{13} + \frac{59960356356944625815}{149529199625785331244} a^{12} - \frac{6310540486034490911}{49843066541928443748} a^{11} + \frac{17641939908368549783}{37382299906446332811} a^{10} + \frac{13571938712567444467}{74764599812892665622} a^{9} - \frac{16954145741161206001}{37382299906446332811} a^{8} - \frac{12003167356478129407}{149529199625785331244} a^{7} - \frac{10947077945112860741}{37382299906446332811} a^{6} - \frac{6146677781229039323}{12460766635482110937} a^{5} + \frac{375644949524864641}{1846039501552905324} a^{4} - \frac{2408008670688233065}{5538118504658715972} a^{3} + \frac{214695012340369226}{461509875388226331} a^{2} - \frac{210251638028826089}{615346500517635108} a + \frac{62166722414447045}{205115500172545036}$, $\frac{1}{897175197754711987464} a^{31} + \frac{1}{448587598877355993732} a^{30} + \frac{1}{112146899719338998433} a^{29} + \frac{289506793907}{897175197754711987464} a^{28} - \frac{117634817917}{897175197754711987464} a^{27} + \frac{1034206781501}{299058399251570662488} a^{26} + \frac{177123129620538185}{897175197754711987464} a^{25} + \frac{74312054083682053}{74764599812892665622} a^{24} + \frac{1147878201452616539}{149529199625785331244} a^{23} - \frac{668557171761868064}{37382299906446332811} a^{22} - \frac{29956757627723690473}{897175197754711987464} a^{21} + \frac{94489818525620023}{112146899719338998433} a^{20} - \frac{17221110157256726389}{448587598877355993732} a^{19} - \frac{21899577362694954049}{897175197754711987464} a^{18} + \frac{66253994441592951965}{897175197754711987464} a^{17} - \frac{2030563744088172391}{149529199625785331244} a^{16} + \frac{25152548344922386787}{299058399251570662488} a^{15} + \frac{41666876581520990039}{149529199625785331244} a^{14} + \frac{143323389634755612863}{897175197754711987464} a^{13} - \frac{4740595067421192727}{99686133083856887496} a^{12} + \frac{262062409140678819221}{897175197754711987464} a^{11} - \frac{288663194492723927389}{897175197754711987464} a^{10} + \frac{30542229215334882469}{448587598877355993732} a^{9} + \frac{23270250202042153933}{448587598877355993732} a^{8} + \frac{11140404622431334175}{112146899719338998433} a^{7} - \frac{38661889405889095303}{299058399251570662488} a^{6} - \frac{15667565447063392985}{49843066541928443748} a^{5} + \frac{2791637822502504103}{16614355513976147916} a^{4} - \frac{1537631671201221221}{11076237009317431944} a^{3} - \frac{502339503269218729}{1230693001035270216} a^{2} + \frac{441667630124167739}{1230693001035270216} a + \frac{40192623830384865}{410231000345090072}$

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

Class group and class number

$C_{104}$, which has order $104$ (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:  \( \frac{47307868738559351}{299058399251570662488} a^{31} - \frac{3425684769569023}{12460766635482110937} a^{30} + \frac{85642119239225575}{149529199625785331244} a^{29} - \frac{763927703613892129}{299058399251570662488} a^{28} + \frac{1517578352919077189}{299058399251570662488} a^{27} - \frac{2154397155295138613}{299058399251570662488} a^{26} + \frac{8266177348970052499}{299058399251570662488} a^{25} - \frac{8656705412700921121}{149529199625785331244} a^{24} + \frac{2093093394206673053}{24921533270964221874} a^{23} - \frac{510427030665784427}{1846039501552905324} a^{22} + \frac{125474429347062377023}{299058399251570662488} a^{21} - \frac{20941210996375437599}{37382299906446332811} a^{20} + \frac{10712116274442334921}{5538118504658715972} a^{19} - \frac{99773068913697794875}{33228711027952295832} a^{18} + \frac{837412067605917241373}{299058399251570662488} a^{17} - \frac{1319126349085521816619}{149529199625785331244} a^{16} + \frac{1334273386584208337293}{99686133083856887496} a^{15} + \frac{7361796569803830427}{923019750776452662} a^{14} + \frac{1516978858084402609975}{299058399251570662488} a^{13} + \frac{3110785548495929698771}{299058399251570662488} a^{12} - \frac{2824060181791403278139}{299058399251570662488} a^{11} + \frac{2727403463194382058749}{299058399251570662488} a^{10} + \frac{204794286894375332986}{37382299906446332811} a^{9} + \frac{130080102070074941356}{37382299906446332811} a^{8} + \frac{222337218599338299769}{49843066541928443748} a^{7} - \frac{1599699254602031439961}{299058399251570662488} a^{6} + \frac{20797332236053538633}{5538118504658715972} a^{5} + \frac{2997474173372895125}{1846039501552905324} a^{4} + \frac{1031131115640275923}{1230693001035270216} a^{3} + \frac{483021552509232243}{410231000345090072} a^{2} - \frac{2460860291101500197}{1230693001035270216} a + \frac{154155814630606035}{410231000345090072} \) (order $30$)
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:  \( 594640437801.8693 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{65})\), 4.4.274625.1, 4.0.2471625.2, \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), 4.4.274625.2, 4.0.2471625.1, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{13})\), 4.4.190125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), 4.0.21125.1, 4.4.19773.1, 4.0.54925.1, 4.4.494325.1, 4.0.2197.1, 8.0.6108930140625.1, 8.0.1445900625.1, 8.0.6108930140625.3, 8.8.75418890625.1, 8.0.6108930140625.2, 8.0.6108930140625.4, 8.0.6108930140625.5, 8.0.36147515625.3, 8.0.36147515625.1, 8.0.244357205625.2, 8.0.244357205625.1, 8.8.36147515625.1, 8.0.446265625.1, 8.8.244357205625.1, 8.0.3016755625.1, 8.0.36147515625.2, \(\Q(\zeta_{15})\), 8.0.390971529.1, 8.0.244357205625.3, 16.0.37319027463036582275390625.1, 16.0.1306642885859619140625.1, 16.0.59710443940858531640625.1, 16.16.37319027463036582275390625.1, 16.0.5688009063105712890625.1, 16.0.37319027463036582275390625.2, 16.0.37319027463036582275390625.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$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}$
5Data not computed
13Data not computed