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Properties

Label 36.0.68583793803...5625.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{48}\cdot 5^{18}\cdot 7^{30}$
Root discriminant $48.97$
Ramified primes $3, 5, 7$
Class number $2268$ (GRH)
Class group $[18, 126]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 18, -77, 372, -1710, 8009, 10350, 30861, 47916, 103692, 148752, 408896, 105954, 415071, 53105, 423768, 62091, 513316, 341952, 443511, 272002, 347325, 164610, 241820, 39264, 34614, 7326, 4881, 981, 584, -69, 72, -4, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 9*x^34 - 4*x^33 + 72*x^32 - 69*x^31 + 584*x^30 + 981*x^29 + 4881*x^28 + 7326*x^27 + 34614*x^26 + 39264*x^25 + 241820*x^24 + 164610*x^23 + 347325*x^22 + 272002*x^21 + 443511*x^20 + 341952*x^19 + 513316*x^18 + 62091*x^17 + 423768*x^16 + 53105*x^15 + 415071*x^14 + 105954*x^13 + 408896*x^12 + 148752*x^11 + 103692*x^10 + 47916*x^9 + 30861*x^8 + 10350*x^7 + 8009*x^6 - 1710*x^5 + 372*x^4 - 77*x^3 + 18*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^36 + 9*x^34 - 4*x^33 + 72*x^32 - 69*x^31 + 584*x^30 + 981*x^29 + 4881*x^28 + 7326*x^27 + 34614*x^26 + 39264*x^25 + 241820*x^24 + 164610*x^23 + 347325*x^22 + 272002*x^21 + 443511*x^20 + 341952*x^19 + 513316*x^18 + 62091*x^17 + 423768*x^16 + 53105*x^15 + 415071*x^14 + 105954*x^13 + 408896*x^12 + 148752*x^11 + 103692*x^10 + 47916*x^9 + 30861*x^8 + 10350*x^7 + 8009*x^6 - 1710*x^5 + 372*x^4 - 77*x^3 + 18*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} + 9 x^{34} - 4 x^{33} + 72 x^{32} - 69 x^{31} + 584 x^{30} + 981 x^{29} + 4881 x^{28} + 7326 x^{27} + 34614 x^{26} + 39264 x^{25} + 241820 x^{24} + 164610 x^{23} + 347325 x^{22} + 272002 x^{21} + 443511 x^{20} + 341952 x^{19} + 513316 x^{18} + 62091 x^{17} + 423768 x^{16} + 53105 x^{15} + 415071 x^{14} + 105954 x^{13} + 408896 x^{12} + 148752 x^{11} + 103692 x^{10} + 47916 x^{9} + 30861 x^{8} + 10350 x^{7} + 8009 x^{6} - 1710 x^{5} + 372 x^{4} - 77 x^{3} + 18 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6858379380370190025774854438611053598470472411685943603515625=3^{48}\cdot 5^{18}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.97$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(4,·)$, $\chi_{315}(136,·)$, $\chi_{315}(139,·)$, $\chi_{315}(271,·)$, $\chi_{315}(16,·)$, $\chi_{315}(274,·)$, $\chi_{315}(19,·)$, $\chi_{315}(151,·)$, $\chi_{315}(286,·)$, $\chi_{315}(31,·)$, $\chi_{315}(289,·)$, $\chi_{315}(34,·)$, $\chi_{315}(166,·)$, $\chi_{315}(169,·)$, $\chi_{315}(46,·)$, $\chi_{315}(304,·)$, $\chi_{315}(181,·)$, $\chi_{315}(184,·)$, $\chi_{315}(61,·)$, $\chi_{315}(64,·)$, $\chi_{315}(199,·)$, $\chi_{315}(76,·)$, $\chi_{315}(79,·)$, $\chi_{315}(211,·)$, $\chi_{315}(214,·)$, $\chi_{315}(94,·)$, $\chi_{315}(226,·)$, $\chi_{315}(229,·)$, $\chi_{315}(106,·)$, $\chi_{315}(109,·)$, $\chi_{315}(241,·)$, $\chi_{315}(244,·)$, $\chi_{315}(121,·)$, $\chi_{315}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{26} a^{21} - \frac{3}{26} a^{14} + \frac{3}{26} a^{7} - \frac{4}{13}$, $\frac{1}{26} a^{22} - \frac{3}{26} a^{15} + \frac{3}{26} a^{8} - \frac{4}{13} a$, $\frac{1}{26} a^{23} - \frac{3}{26} a^{16} + \frac{3}{26} a^{9} - \frac{4}{13} a^{2}$, $\frac{1}{26} a^{24} - \frac{3}{26} a^{17} + \frac{3}{26} a^{10} - \frac{4}{13} a^{3}$, $\frac{1}{26} a^{25} - \frac{3}{26} a^{18} + \frac{3}{26} a^{11} - \frac{4}{13} a^{4}$, $\frac{1}{60814} a^{26} + \frac{775}{60814} a^{25} - \frac{498}{30407} a^{24} - \frac{45}{60814} a^{23} + \frac{175}{30407} a^{22} - \frac{527}{30407} a^{21} + \frac{225}{4678} a^{20} + \frac{4646}{30407} a^{19} - \frac{2273}{60814} a^{18} + \frac{5835}{60814} a^{17} - \frac{4320}{30407} a^{16} - \frac{9591}{60814} a^{15} + \frac{3955}{60814} a^{14} + \frac{1885}{4678} a^{13} - \frac{9300}{30407} a^{12} - \frac{12365}{60814} a^{11} - \frac{25361}{60814} a^{10} + \frac{5360}{30407} a^{9} - \frac{26679}{60814} a^{8} - \frac{19139}{60814} a^{7} + \frac{501}{4678} a^{6} + \frac{30399}{60814} a^{5} - \frac{13266}{30407} a^{4} - \frac{28913}{60814} a^{3} + \frac{30377}{60814} a^{2} + \frac{30337}{60814} a - \frac{22937}{60814}$, $\frac{1}{60814} a^{27} - \frac{249}{30407} a^{25} - \frac{15}{60814} a^{24} + \frac{70}{30407} a^{23} - \frac{490}{30407} a^{22} + \frac{1125}{60814} a^{21} - \frac{7465}{60814} a^{20} + \frac{111}{2339} a^{19} - \frac{2776}{30407} a^{18} + \frac{2278}{30407} a^{17} + \frac{10903}{60814} a^{16} - \frac{2859}{30407} a^{15} - \frac{6937}{60814} a^{14} + \frac{24807}{60814} a^{13} + \frac{774}{2339} a^{12} + \frac{9536}{30407} a^{11} - \frac{6256}{30407} a^{10} - \frac{17195}{60814} a^{9} + \frac{1858}{30407} a^{8} + \frac{7639}{60814} a^{7} + \frac{15193}{30407} a^{6} - \frac{782}{2339} a^{5} + \frac{1992}{30407} a^{4} - \frac{75}{60814} a^{3} + \frac{30197}{60814} a^{2} + \frac{9809}{30407} a + \frac{15091}{30407}$, $\frac{1}{121628} a^{28} - \frac{27}{2339} a^{21} + \frac{6247}{121628} a^{14} + \frac{14125}{60814} a^{7} - \frac{10645}{121628}$, $\frac{1}{121628} a^{29} - \frac{27}{2339} a^{22} + \frac{6247}{121628} a^{15} + \frac{14125}{60814} a^{8} - \frac{10645}{121628} a$, $\frac{1}{2310932} a^{30} - \frac{7}{2310932} a^{29} + \frac{9}{2310932} a^{28} - \frac{5}{1155466} a^{27} - \frac{7}{1155466} a^{26} - \frac{2637}{577733} a^{25} + \frac{15}{577733} a^{24} + \frac{12947}{1155466} a^{23} - \frac{3335}{577733} a^{22} - \frac{13921}{1155466} a^{21} - \frac{256813}{1155466} a^{20} - \frac{70144}{577733} a^{19} + \frac{25344}{577733} a^{18} + \frac{109461}{1155466} a^{17} + \frac{4277}{177764} a^{16} + \frac{110301}{2310932} a^{15} - \frac{420967}{2310932} a^{14} - \frac{143535}{1155466} a^{13} - \frac{15617}{577733} a^{12} - \frac{68725}{577733} a^{11} - \frac{358697}{1155466} a^{10} + \frac{216023}{577733} a^{9} + \frac{17419}{44441} a^{8} - \frac{282670}{577733} a^{7} - \frac{53150}{577733} a^{6} + \frac{253751}{1155466} a^{5} + \frac{183479}{577733} a^{4} + \frac{410937}{1155466} a^{3} + \frac{132215}{2310932} a^{2} - \frac{1051607}{2310932} a - \frac{1082277}{2310932}$, $\frac{1}{459502349931691508} a^{31} + \frac{23105260015}{114875587482922877} a^{30} - \frac{1111292257065}{459502349931691508} a^{29} + \frac{207947340134}{114875587482922877} a^{28} - \frac{703862467853}{114875587482922877} a^{27} - \frac{117156495543}{229751174965845754} a^{26} - \frac{42927993103091}{8836583652532529} a^{25} + \frac{3147413415081757}{229751174965845754} a^{24} + \frac{175977625160275}{229751174965845754} a^{23} + \frac{168219920864031}{114875587482922877} a^{22} - \frac{481231921091797}{114875587482922877} a^{21} + \frac{27200695337466463}{229751174965845754} a^{20} + \frac{11181936291486630}{114875587482922877} a^{19} - \frac{2659256307080921}{17673167305065058} a^{18} - \frac{58579352787510683}{459502349931691508} a^{17} - \frac{11492804326892879}{229751174965845754} a^{16} + \frac{46070785904281803}{459502349931691508} a^{15} - \frac{2123872751781687}{12092167103465566} a^{14} - \frac{71044558038831579}{229751174965845754} a^{13} + \frac{5814693138203683}{114875587482922877} a^{12} + \frac{2505639853705799}{17673167305065058} a^{11} - \frac{203517809502569}{6046083551732783} a^{10} + \frac{54856969399265275}{229751174965845754} a^{9} - \frac{2985812364346790}{8836583652532529} a^{8} - \frac{34420005421255755}{229751174965845754} a^{7} - \frac{102233932772235345}{229751174965845754} a^{6} - \frac{16288441611747155}{229751174965845754} a^{5} - \frac{5116460069069085}{17673167305065058} a^{4} - \frac{220772918270385917}{459502349931691508} a^{3} - \frac{2774257102949717}{8836583652532529} a^{2} - \frac{41439766729129861}{459502349931691508} a + \frac{10145069958298437}{229751174965845754}$, $\frac{1}{459502349931691508} a^{32} - \frac{23764359469}{459502349931691508} a^{30} - \frac{15135525266}{8836583652532529} a^{29} - \frac{106939617615}{229751174965845754} a^{28} - \frac{1605218042699}{229751174965845754} a^{27} + \frac{359288843370}{114875587482922877} a^{26} - \frac{1036040538561980}{114875587482922877} a^{25} - \frac{4364076496193435}{229751174965845754} a^{24} + \frac{3352969581890193}{229751174965845754} a^{23} + \frac{1311132126040017}{229751174965845754} a^{22} - \frac{169629107282369}{8836583652532529} a^{21} - \frac{50731886729387727}{229751174965845754} a^{20} + \frac{25553281014785024}{114875587482922877} a^{19} - \frac{44262898417200043}{459502349931691508} a^{18} - \frac{1193697249771950}{114875587482922877} a^{17} + \frac{22726752930241405}{459502349931691508} a^{16} - \frac{11454686829293095}{229751174965845754} a^{15} + \frac{25645439333575422}{114875587482922877} a^{14} - \frac{6834379746167617}{229751174965845754} a^{13} - \frac{24141690446674730}{114875587482922877} a^{12} - \frac{2011344273427917}{114875587482922877} a^{11} + \frac{17469369312965688}{114875587482922877} a^{10} - \frac{42006522080644317}{229751174965845754} a^{9} + \frac{58093365258248825}{229751174965845754} a^{8} - \frac{41242874014358405}{229751174965845754} a^{7} + \frac{7983114566410781}{114875587482922877} a^{6} + \frac{34848869625115920}{114875587482922877} a^{5} + \frac{54177157081791569}{459502349931691508} a^{4} + \frac{102413395232574347}{229751174965845754} a^{3} - \frac{35589419847258733}{459502349931691508} a^{2} + \frac{23782168148075253}{114875587482922877} a + \frac{45936605508749034}{114875587482922877}$, $\frac{1}{459502349931691508} a^{33} - \frac{18297050831}{114875587482922877} a^{30} + \frac{58351001975}{17673167305065058} a^{29} - \frac{658693829913}{459502349931691508} a^{28} + \frac{100503475839}{17673167305065058} a^{27} - \frac{707998243103}{229751174965845754} a^{26} + \frac{186753388451783}{17673167305065058} a^{25} - \frac{2159130434410549}{229751174965845754} a^{24} - \frac{2310724442254107}{229751174965845754} a^{23} - \frac{1512500267865529}{114875587482922877} a^{22} + \frac{667686234541668}{114875587482922877} a^{21} - \frac{1057455839530923}{8836583652532529} a^{20} - \frac{103488695567283465}{459502349931691508} a^{19} + \frac{1590895333088961}{8836583652532529} a^{18} + \frac{29700342546820991}{229751174965845754} a^{17} - \frac{44906078963075479}{229751174965845754} a^{16} - \frac{7316709780696113}{114875587482922877} a^{15} - \frac{70471225955341845}{459502349931691508} a^{14} - \frac{1918708260185752}{8836583652532529} a^{13} - \frac{47420508877578945}{229751174965845754} a^{12} + \frac{1068197436601199}{8836583652532529} a^{11} + \frac{11045006387640609}{229751174965845754} a^{10} + \frac{40125056471527363}{229751174965845754} a^{9} - \frac{45481389679474063}{229751174965845754} a^{8} - \frac{9560827519336387}{114875587482922877} a^{7} - \frac{536371961634459}{17673167305065058} a^{6} + \frac{15376415952699381}{35346334610130116} a^{5} - \frac{5692438482431513}{17673167305065058} a^{4} + \frac{20130486705499750}{114875587482922877} a^{3} - \frac{14907190834978446}{114875587482922877} a^{2} - \frac{15466056885849033}{114875587482922877} a + \frac{33974795000760547}{459502349931691508}$, $\frac{1}{459502349931691508} a^{34} + \frac{4514409845}{459502349931691508} a^{30} - \frac{204492063713}{114875587482922877} a^{29} + \frac{1562680331}{17673167305065058} a^{28} - \frac{308911468941}{114875587482922877} a^{27} - \frac{90510886477}{229751174965845754} a^{26} - \frac{790221906980721}{114875587482922877} a^{25} - \frac{1576083115547505}{229751174965845754} a^{24} + \frac{2484635217579643}{229751174965845754} a^{23} + \frac{2700659675777107}{229751174965845754} a^{22} - \frac{1293763872200173}{229751174965845754} a^{21} + \frac{79784419458856239}{459502349931691508} a^{20} - \frac{13118480586150491}{229751174965845754} a^{19} + \frac{36729832056917327}{229751174965845754} a^{18} - \frac{37252655867562143}{229751174965845754} a^{17} + \frac{97504498546131031}{459502349931691508} a^{16} - \frac{1358484038407810}{114875587482922877} a^{15} + \frac{45671138725778219}{229751174965845754} a^{14} - \frac{95860099268741653}{229751174965845754} a^{13} + \frac{20021541428530557}{229751174965845754} a^{12} - \frac{107046380783247389}{229751174965845754} a^{11} - \frac{26199770755753639}{229751174965845754} a^{10} + \frac{52155585062049559}{229751174965845754} a^{9} + \frac{27725372212495619}{114875587482922877} a^{8} + \frac{2780028514405198}{6046083551732783} a^{7} + \frac{152145393730595067}{459502349931691508} a^{6} - \frac{51577324636201177}{114875587482922877} a^{5} + \frac{24583825859839657}{229751174965845754} a^{4} - \frac{11267363501960789}{114875587482922877} a^{3} + \frac{16894092421857561}{35346334610130116} a^{2} - \frac{106263534103355287}{229751174965845754} a - \frac{28309655120558772}{114875587482922877}$, $\frac{1}{459502349931691508} a^{35} - \frac{705869272569}{229751174965845754} a^{28} + \frac{7425074152862345}{459502349931691508} a^{21} - \frac{49573374082423977}{229751174965845754} a^{14} - \frac{158073826101113367}{459502349931691508} a^{7} + \frac{25303216547920385}{229751174965845754}$

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

Class group and class number

$C_{18}\times C_{126}$, which has order $2268$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{54061831489131}{6046083551732783} a^{35} - \frac{486556483402179}{24184334206931132} a^{34} - \frac{1784040439141323}{24184334206931132} a^{33} - \frac{864989303826096}{6046083551732783} a^{32} - \frac{233922457508511}{465083350133291} a^{31} - \frac{20273186808424125}{24184334206931132} a^{30} - \frac{81254932728163893}{24184334206931132} a^{29} - \frac{503585960321255265}{24184334206931132} a^{28} - \frac{720914522907561885}{12092167103465566} a^{27} - \frac{144939770222360211}{930166700266582} a^{26} - \frac{5102517841438651173}{12092167103465566} a^{25} - \frac{11940228078317456977}{12092167103465566} a^{24} - \frac{16346784111031518732}{6046083551732783} a^{23} - \frac{72508106826042921117}{12092167103465566} a^{22} - \frac{56928892598494084323}{12092167103465566} a^{21} - \frac{188042349417509042073}{24184334206931132} a^{20} - \frac{163022209433350280487}{24184334206931132} a^{19} - \frac{56711888406896712489}{6046083551732783} a^{18} - \frac{47667225350402219025}{6046083551732783} a^{17} - \frac{183747569400349497171}{24184334206931132} a^{16} - \frac{20110514757473329821}{24184334206931132} a^{15} - \frac{179849819473646130333}{24184334206931132} a^{14} - \frac{21755400042361629627}{12092167103465566} a^{13} - \frac{48336305101611603183}{6046083551732783} a^{12} - \frac{17446455825351932403}{6046083551732783} a^{11} - \frac{47918490775391271702}{6046083551732783} a^{10} - \frac{11377150251393151557}{12092167103465566} a^{9} - \frac{3645659606469548985}{6046083551732783} a^{8} - \frac{1282076333764741665}{6046083551732783} a^{7} - \frac{3876719874254094879}{24184334206931132} a^{6} + \frac{827848825593063003}{24184334206931132} a^{5} - \frac{90012949429403115}{12092167103465566} a^{4} + \frac{1239982659569615589}{12092167103465566} a^{3} - \frac{8703954869750091}{24184334206931132} a^{2} + \frac{1459669450206537}{24184334206931132} a - \frac{486556483402179}{24184334206931132} \) (order $14$)
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:  \( 13624539961495.691 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(\sqrt{5}, \sqrt{-7})\), 6.0.281302875.3, 6.0.13783840875.2, 6.0.2100875.1, 6.0.13783840875.1, 6.6.820125.1, 6.0.2250423.1, 6.6.1969120125.2, 6.0.110270727.1, 6.6.300125.1, \(\Q(\zeta_{7})\), 6.6.1969120125.1, 6.0.110270727.2, 9.9.62523502209.1, 12.0.79131307483265625.1, 12.0.189994269267320765625.2, 12.0.4413675765625.1, 12.0.189994269267320765625.1, 18.0.2618850774742652270958169921875.4, 18.18.7635133454060210702501953125.1, 18.0.1340851596668237962730583.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
3Data not computed
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed