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Properties

Label 36.0.86695216288...8125.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 5^{27}\cdot 19^{34}$
Root discriminant $93.43$
Ramified primes $3, 5, 19$
Class number Not computed
Class group Not computed
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32387434201, 22060436420, 54447870260, 17339912395, 71787766410, -10313272211, 61474246192, -24802862407, 36669453879, -19560482921, 17099968701, -10088647824, 6983906898, -4214492261, 2712046322, -1592813131, 1034017336, -560976311, 381365075, -179361007, 129770818, -49590189, 38325434, -11264755, 9259294, -2005461, 1738095, -267366, 241980, -25386, 23782, -1604, 1544, -60, 59, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 59*x^34 - 60*x^33 + 1544*x^32 - 1604*x^31 + 23782*x^30 - 25386*x^29 + 241980*x^28 - 267366*x^27 + 1738095*x^26 - 2005461*x^25 + 9259294*x^24 - 11264755*x^23 + 38325434*x^22 - 49590189*x^21 + 129770818*x^20 - 179361007*x^19 + 381365075*x^18 - 560976311*x^17 + 1034017336*x^16 - 1592813131*x^15 + 2712046322*x^14 - 4214492261*x^13 + 6983906898*x^12 - 10088647824*x^11 + 17099968701*x^10 - 19560482921*x^9 + 36669453879*x^8 - 24802862407*x^7 + 61474246192*x^6 - 10313272211*x^5 + 71787766410*x^4 + 17339912395*x^3 + 54447870260*x^2 + 22060436420*x + 32387434201)
 
gp: K = bnfinit(x^36 - x^35 + 59*x^34 - 60*x^33 + 1544*x^32 - 1604*x^31 + 23782*x^30 - 25386*x^29 + 241980*x^28 - 267366*x^27 + 1738095*x^26 - 2005461*x^25 + 9259294*x^24 - 11264755*x^23 + 38325434*x^22 - 49590189*x^21 + 129770818*x^20 - 179361007*x^19 + 381365075*x^18 - 560976311*x^17 + 1034017336*x^16 - 1592813131*x^15 + 2712046322*x^14 - 4214492261*x^13 + 6983906898*x^12 - 10088647824*x^11 + 17099968701*x^10 - 19560482921*x^9 + 36669453879*x^8 - 24802862407*x^7 + 61474246192*x^6 - 10313272211*x^5 + 71787766410*x^4 + 17339912395*x^3 + 54447870260*x^2 + 22060436420*x + 32387434201, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 59 x^{34} - 60 x^{33} + 1544 x^{32} - 1604 x^{31} + 23782 x^{30} - 25386 x^{29} + 241980 x^{28} - 267366 x^{27} + 1738095 x^{26} - 2005461 x^{25} + 9259294 x^{24} - 11264755 x^{23} + 38325434 x^{22} - 49590189 x^{21} + 129770818 x^{20} - 179361007 x^{19} + 381365075 x^{18} - 560976311 x^{17} + 1034017336 x^{16} - 1592813131 x^{15} + 2712046322 x^{14} - 4214492261 x^{13} + 6983906898 x^{12} - 10088647824 x^{11} + 17099968701 x^{10} - 19560482921 x^{9} + 36669453879 x^{8} - 24802862407 x^{7} + 61474246192 x^{6} - 10313272211 x^{5} + 71787766410 x^{4} + 17339912395 x^{3} + 54447870260 x^{2} + 22060436420 x + 32387434201 \)

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:  \(86695216288678587726682989606644567242259916538384325802326202392578125=3^{18}\cdot 5^{27}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.43$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(285=3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{285}(256,·)$, $\chi_{285}(1,·)$, $\chi_{285}(2,·)$, $\chi_{285}(4,·)$, $\chi_{285}(257,·)$, $\chi_{285}(8,·)$, $\chi_{285}(128,·)$, $\chi_{285}(139,·)$, $\chi_{285}(271,·)$, $\chi_{285}(16,·)$, $\chi_{285}(278,·)$, $\chi_{285}(32,·)$, $\chi_{285}(167,·)$, $\chi_{285}(169,·)$, $\chi_{285}(173,·)$, $\chi_{285}(49,·)$, $\chi_{285}(53,·)$, $\chi_{285}(61,·)$, $\chi_{285}(64,·)$, $\chi_{285}(196,·)$, $\chi_{285}(199,·)$, $\chi_{285}(203,·)$, $\chi_{285}(98,·)$, $\chi_{285}(212,·)$, $\chi_{285}(214,·)$, $\chi_{285}(143,·)$, $\chi_{285}(226,·)$, $\chi_{285}(227,·)$, $\chi_{285}(229,·)$, $\chi_{285}(106,·)$, $\chi_{285}(107,·)$, $\chi_{285}(113,·)$, $\chi_{285}(242,·)$, $\chi_{285}(244,·)$, $\chi_{285}(121,·)$, $\chi_{285}(122,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{4835148121} a^{28} + \frac{2296583799}{4835148121} a^{27} - \frac{481936019}{4835148121} a^{26} + \frac{1274714130}{4835148121} a^{25} - \frac{1732083832}{4835148121} a^{24} + \frac{206753914}{4835148121} a^{23} + \frac{208165548}{4835148121} a^{22} + \frac{2264424117}{4835148121} a^{21} - \frac{340367871}{4835148121} a^{20} + \frac{2375167882}{4835148121} a^{19} + \frac{1576508306}{4835148121} a^{18} + \frac{1874617694}{4835148121} a^{17} - \frac{323757407}{4835148121} a^{16} - \frac{1473007474}{4835148121} a^{15} + \frac{1915397156}{4835148121} a^{14} - \frac{340507754}{4835148121} a^{13} + \frac{928009878}{4835148121} a^{12} - \frac{1563187271}{4835148121} a^{11} - \frac{411294693}{4835148121} a^{10} - \frac{684604269}{4835148121} a^{9} - \frac{375094073}{4835148121} a^{8} + \frac{440765388}{4835148121} a^{7} - \frac{429984318}{4835148121} a^{6} - \frac{449557576}{4835148121} a^{5} - \frac{2059096911}{4835148121} a^{4} - \frac{755941256}{4835148121} a^{3} + \frac{1633029567}{4835148121} a^{2} + \frac{1556782435}{4835148121} a - \frac{144074371}{4835148121}$, $\frac{1}{4835148121} a^{29} + \frac{372449937}{4835148121} a^{27} - \frac{639776979}{4835148121} a^{26} - \frac{1900279781}{4835148121} a^{25} - \frac{432727015}{4835148121} a^{24} + \frac{977702281}{4835148121} a^{23} + \frac{228536937}{4835148121} a^{22} + \frac{267086056}{4835148121} a^{21} + \frac{543068605}{4835148121} a^{20} + \frac{972871691}{4835148121} a^{19} - \frac{632995209}{4835148121} a^{18} + \frac{1033165600}{4835148121} a^{17} + \frac{772284661}{4835148121} a^{16} - \frac{1317901483}{4835148121} a^{15} - \frac{1739191222}{4835148121} a^{14} + \frac{51158757}{4835148121} a^{13} + \frac{205411208}{4835148121} a^{12} + \frac{758227509}{4835148121} a^{11} + \frac{2308488269}{4835148121} a^{10} + \frac{1128488786}{4835148121} a^{9} + \frac{2153564085}{4835148121} a^{8} + \frac{612746860}{4835148121} a^{7} + \frac{901918276}{4835148121} a^{6} - \frac{959300283}{4835148121} a^{5} + \frac{1255205229}{4835148121} a^{4} - \frac{2136360265}{4835148121} a^{3} - \frac{437419312}{4835148121} a^{2} + \frac{258103362}{4835148121} a + \frac{403147187}{4835148121}$, $\frac{1}{4835148121} a^{30} + \frac{516768725}{4835148121} a^{27} + \frac{1797280163}{4835148121} a^{26} - \frac{340602637}{4835148121} a^{25} + \frac{1383718353}{4835148121} a^{24} + \frac{802774388}{4835148121} a^{23} + \frac{2183337747}{4835148121} a^{22} - \frac{2283441028}{4835148121} a^{21} + \frac{533575788}{4835148121} a^{20} - \frac{272116129}{4835148121} a^{19} - \frac{477696624}{4835148121} a^{18} - \frac{2055474207}{4835148121} a^{17} - \frac{841866775}{4835148121} a^{16} + \frac{2223788495}{4835148121} a^{15} + \frac{1502734825}{4835148121} a^{14} + \frac{459648901}{4835148121} a^{13} - \frac{2331064909}{4835148121} a^{12} - \frac{1648142508}{4835148121} a^{11} - \frac{465084894}{4835148121} a^{10} - \frac{1546467954}{4835148121} a^{9} + \frac{2124627523}{4835148121} a^{8} + \frac{781544019}{4835148121} a^{7} + \frac{1932791407}{4835148121} a^{6} - \frac{1174156818}{4835148121} a^{5} + \frac{625754436}{4835148121} a^{4} - \frac{1033054397}{4835148121} a^{3} - \frac{934937804}{4835148121} a^{2} + \frac{1942052373}{4835148121} a + \frac{2049962264}{4835148121}$, $\frac{1}{4835148121} a^{31} + \frac{443941629}{4835148121} a^{27} - \frac{431861197}{4835148121} a^{26} - \frac{216594750}{4835148121} a^{25} + \frac{47365133}{4835148121} a^{24} + \frac{1153229326}{4835148121} a^{23} + \frac{1686563534}{4835148121} a^{22} + \frac{2185131917}{4835148121} a^{21} - \frac{642820934}{4835148121} a^{20} + \frac{1631376421}{4835148121} a^{19} - \frac{689063611}{4835148121} a^{18} - \frac{1015385642}{4835148121} a^{17} - \frac{517865225}{4835148121} a^{16} - \frac{160991377}{4835148121} a^{15} - \frac{2046044548}{4835148121} a^{14} - \frac{1006619489}{4835148121} a^{13} + \frac{241412673}{4835148121} a^{12} + \frac{978704255}{4835148121} a^{11} - \frac{1512801373}{4835148121} a^{10} - \frac{358324551}{4835148121} a^{9} + \frac{213638093}{4835148121} a^{8} + \frac{388971306}{4835148121} a^{7} - \frac{665357733}{4835148121} a^{6} - \frac{715231411}{4835148121} a^{5} + \frac{1303602031}{4835148121} a^{4} - \frac{957609144}{4835148121} a^{3} + \frac{2358287957}{4835148121} a^{2} + \frac{1091014722}{4835148121} a + \frac{3178981}{4835148121}$, $\frac{1}{4835148121} a^{32} - \frac{315036017}{4835148121} a^{27} + \frac{720546396}{4835148121} a^{26} - \frac{238865539}{4835148121} a^{25} + \frac{1801226600}{4835148121} a^{24} - \frac{1688637115}{4835148121} a^{23} - \frac{739617708}{4835148121} a^{22} - \frac{1206261494}{4835148121} a^{21} - \frac{1449843133}{4835148121} a^{20} - \frac{1934461840}{4835148121} a^{19} + \frac{739639051}{4835148121} a^{18} + \frac{1918209644}{4835148121} a^{17} - \frac{124627808}{4835148121} a^{16} - \frac{1365659626}{4835148121} a^{15} - \frac{622845591}{4835148121} a^{14} - \frac{760235477}{4835148121} a^{13} - \frac{1889180460}{4835148121} a^{12} - \frac{1952155643}{4835148121} a^{11} + \frac{898018911}{4835148121} a^{10} - \frac{2031688522}{4835148121} a^{9} - \frac{1616974074}{4835148121} a^{8} + \frac{2308110678}{4835148121} a^{7} - \frac{1396328582}{4835148121} a^{6} - \frac{243682467}{4835148121} a^{5} + \frac{746443946}{4835148121} a^{4} + \frac{2090546283}{4835148121} a^{3} - \frac{1277357866}{4835148121} a^{2} + \frac{2060371472}{4835148121} a - \frac{29653464}{4835148121}$, $\frac{1}{4835148121} a^{33} - \frac{1997900661}{4835148121} a^{27} + \frac{719807073}{4835148121} a^{26} + \frac{1327822979}{4835148121} a^{25} + \frac{936858519}{4835148121} a^{24} + \frac{578263616}{4835148121} a^{23} - \frac{1919135488}{4835148121} a^{22} + \frac{705927954}{4835148121} a^{21} + \frac{1531609879}{4835148121} a^{20} + \frac{328371988}{4835148121} a^{19} - \frac{9751026}{4835148121} a^{18} + \frac{2094475362}{4835148121} a^{17} - \frac{556235721}{4835148121} a^{16} - \frac{2264971096}{4835148121} a^{15} - \frac{2320190816}{4835148121} a^{14} + \frac{933700042}{4835148121} a^{13} - \frac{1125315293}{4835148121} a^{12} - \frac{155147322}{4835148121} a^{11} - \frac{1285571712}{4835148121} a^{10} - \frac{1598077061}{4835148121} a^{9} - \frac{2121156679}{4835148121} a^{8} - \frac{2314664752}{4835148121} a^{7} + \frac{404596048}{4835148121} a^{6} - \frac{941155262}{4835148121} a^{5} - \frac{2289201719}{4835148121} a^{4} - \frac{1392794842}{4835148121} a^{3} + \frac{2318645774}{4835148121} a^{2} - \frac{2172239054}{4835148121} a - \frac{2341762324}{4835148121}$, $\frac{1}{4835148121} a^{34} - \frac{1164228666}{4835148121} a^{27} + \frac{888720483}{4835148121} a^{26} + \frac{443016248}{4835148121} a^{25} + \frac{60513117}{4835148121} a^{24} + \frac{1727070643}{4835148121} a^{23} + \frac{800881706}{4835148121} a^{22} + \frac{1022688475}{4835148121} a^{21} - \frac{650617550}{4835148121} a^{20} + \frac{2095525752}{4835148121} a^{19} + \frac{2361423824}{4835148121} a^{18} + \frac{1894953334}{4835148121} a^{17} + \frac{127228207}{4835148121} a^{16} - \frac{293802283}{4835148121} a^{15} - \frac{1771819383}{4835148121} a^{14} + \frac{1954074427}{4835148121} a^{13} + \frac{756545164}{4835148121} a^{12} + \frac{2111911782}{4835148121} a^{11} - \frac{2253910442}{4835148121} a^{10} - \frac{809594812}{4835148121} a^{9} + \frac{1318718341}{4835148121} a^{8} - \frac{977310334}{4835148121} a^{7} + \frac{679014373}{4835148121} a^{6} - \frac{65870961}{4835148121} a^{5} - \frac{1573629375}{4835148121} a^{4} + \frac{767439934}{4835148121} a^{3} - \frac{1342538540}{4835148121} a^{2} + \frac{2190428449}{4835148121} a - \frac{1357281181}{4835148121}$, $\frac{1}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{35} + \frac{32281992524075341476745873231483000034774830433125937178588409794877403354388924363353063706594807786823453015}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{34} - \frac{595940437584592029793090144551628067949650093841198957069844231751639063158647481650922676577021217300635694064}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{33} - \frac{3060057250598586163255097794381314203760554769531781173850866565447296192398636416235704221866372282010395817638}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{32} - \frac{1601876736368121670629522521939385011921345719902965656428309279186366586856319769184889572457261673460415595777}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{31} - \frac{2418335461669774818403422420383517290173976607976557901216863807184834897311847031435617535013655440835526631344}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{30} + \frac{859852900189766363565821368210978751321350616191735703808335885694522091266707188018836588338243828780241667535}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{29} + \frac{1886839677357189215116862921770254620998668891101251330619830701648242258875649686458637158184627783459740602514}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{28} + \frac{8674117788884484716323448516361514136868827780368879127870869680226541602511837001368079877252110929253299179064714539861}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{27} + \frac{8610883817803232891850041068965839159191125711977411908756741625183766327696728416589083600858927696265172256665471691219}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{26} + \frac{9409468820003162984884306502938177102643545091551420938269732585087733433620379079599365577778418252285679189422243688936}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{25} - \frac{11268241217147435556031816026320217164620555754962620729382094965163042465506358922105483038046888083850313090842392391996}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{24} + \frac{3135990735118521892694094261072956646494514395416947761148988396010383659702840365683540614289559824430024685075542148158}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{23} - \frac{1267451542197590572239699440678880402013805914595653036078128734467352206327838975911241334304250843361405001144964273361}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{22} - \frac{4048208626380536618900074892821766851563919200342511932935343241881599798374919558026879959268279060793265894187009618420}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{21} - \frac{5873282962224259218617599953652615518457797905629318954930164255791916925025891083007251003503646445376246637939045959879}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{20} - \frac{4812327857437424430681946535582887881028997807380210646748979507290009128236825940506142514531802116801495164607808187829}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{19} - \frac{6915162540155513340761681219046708727400995441014380556008891347085974130918354512888376039880236229401158619180649090690}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{18} + \frac{12557023102344231945013221250360246269911637994976906960692996814828334811104159601490043087352856520421052569347626231848}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{17} + \frac{4131607966043645257489675766056458276089080502408384968852355403446285399509359466251586181002661564664241061560376747364}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{16} - \frac{4035397573370466625079651010710036155253841002259148036550723782959234293586616492649806989233107448068382963951941177464}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{15} - \frac{11407630403767298027361014947212337452483257041998729620025228062367681093995957481339791722180097196079975021428868816397}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{14} + \frac{7480535303016442414330125172354959854358091116098030352418419285657889922151833015158163778027244011131937726514657109215}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{13} + \frac{6182150992612657874640062026074773439334120936137880378798733601830666461498607500812645129360275231182670369778867627907}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{12} + \frac{4874088375501566063690574044639226030285786834402287508047160774109879501213216418289890807209884997689591137530035680062}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{11} + \frac{14809877127535854904210904256397774630401796028778022010307180294063734132113393480309634613624630035374275437977919288100}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{10} - \frac{14671269902751521282650966067159578301854247405954954074665014648687818611634445569231060129552998644558715565556736870089}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{9} + \frac{1589683745459655468497690343346823494446770442386264464792180634365684381801714362842474939185012220886021539914014989807}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{8} + \frac{4512148786125167981072926411266664370194349650094234146441550888037620370821982063953083777612490712826954750935670447815}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{7} - \frac{11678312426527665818122190565997039049917704024727576083733989856395258739329043647216737621913548569798573198459417143083}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{6} - \frac{3112844609746854119082493318967511091693354061453324942569279201838013587372666548177419949453467690479334382191548093199}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{5} + \frac{5953074478629730986179245737552867228549579970527112875968726769149550035948857112143934488238926202969754276650088831515}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{4} - \frac{1924743893197527725003995092969362286812353979942817023593368875443807233142310839415782207151214978130883563996566842889}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{3} + \frac{649717438688020603726949464649758570717867024507358706259284834795986661249812065053560599772072694527700460282617987835}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a^{2} + \frac{10684576692852047092100825526442824385878266858961009201836368819033102248302319887683893786019821320353330134465322254333}{30457299969280535694883963143296961819091436393054738571546903072673913833965769062740472650977729772243985760183494645089} a + \frac{37177444889646305248308606379718024011100958095308950258413737814213708243211680312025894471883802606655815514}{940404842824509107055459615144243263455157961528165161351952292390051742400011338976042174776589360001459961289}$

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

Class group and class number

Not computed

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.361.1, 4.0.406125.2, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.0.8729584574095564453125.1, 18.18.563362135874260093126953125.1

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 $36$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $36$ $36$ R $36$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ $36$ $36$ $36$ $18^{2}$

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
5Data not computed
19Data not computed