Properties

Label 45.45.940...689.1
Degree $45$
Signature $[45, 0]$
Discriminant $9.408\times 10^{89}$
Root discriminant $99.86$
Ramified primes $3, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{45}$ (as 45T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 81*x^43 + 2943*x^41 - 63576*x^39 + 912699*x^37 - 256*x^36 - 9221688*x^35 + 11619*x^34 + 67767147*x^33 - 228393*x^32 - 369266337*x^31 + 2577030*x^30 + 1508018391*x^29 - 18638901*x^28 - 4636251652*x^27 + 91308222*x^26 + 10725370218*x^25 - 311639850*x^24 - 18578308230*x^23 + 749536389*x^22 + 23873708220*x^21 - 1268865972*x^20 - 22446274041*x^19 + 1494154410*x^18 + 15154469253*x^17 - 1197712170*x^16 - 7173383040*x^15 + 633986568*x^14 + 2312491545*x^13 - 213869688*x^12 - 490418469*x^11 + 44511588*x^10 + 65361426*x^9 - 5507676*x^8 - 5123475*x^7 + 385695*x^6 + 214380*x^5 - 14391*x^4 - 4176*x^3 + 243*x^2 + 27*x - 1)
 
gp: K = bnfinit(x^45 - 81*x^43 + 2943*x^41 - 63576*x^39 + 912699*x^37 - 256*x^36 - 9221688*x^35 + 11619*x^34 + 67767147*x^33 - 228393*x^32 - 369266337*x^31 + 2577030*x^30 + 1508018391*x^29 - 18638901*x^28 - 4636251652*x^27 + 91308222*x^26 + 10725370218*x^25 - 311639850*x^24 - 18578308230*x^23 + 749536389*x^22 + 23873708220*x^21 - 1268865972*x^20 - 22446274041*x^19 + 1494154410*x^18 + 15154469253*x^17 - 1197712170*x^16 - 7173383040*x^15 + 633986568*x^14 + 2312491545*x^13 - 213869688*x^12 - 490418469*x^11 + 44511588*x^10 + 65361426*x^9 - 5507676*x^8 - 5123475*x^7 + 385695*x^6 + 214380*x^5 - 14391*x^4 - 4176*x^3 + 243*x^2 + 27*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 27, 243, -4176, -14391, 214380, 385695, -5123475, -5507676, 65361426, 44511588, -490418469, -213869688, 2312491545, 633986568, -7173383040, -1197712170, 15154469253, 1494154410, -22446274041, -1268865972, 23873708220, 749536389, -18578308230, -311639850, 10725370218, 91308222, -4636251652, -18638901, 1508018391, 2577030, -369266337, -228393, 67767147, 11619, -9221688, -256, 912699, 0, -63576, 0, 2943, 0, -81, 0, 1]);
 

\( x^{45} - 81 x^{43} + 2943 x^{41} - 63576 x^{39} + 912699 x^{37} - 256 x^{36} - 9221688 x^{35} + 11619 x^{34} + 67767147 x^{33} - 228393 x^{32} - 369266337 x^{31} + 2577030 x^{30} + 1508018391 x^{29} - 18638901 x^{28} - 4636251652 x^{27} + 91308222 x^{26} + 10725370218 x^{25} - 311639850 x^{24} - 18578308230 x^{23} + 749536389 x^{22} + 23873708220 x^{21} - 1268865972 x^{20} - 22446274041 x^{19} + 1494154410 x^{18} + 15154469253 x^{17} - 1197712170 x^{16} - 7173383040 x^{15} + 633986568 x^{14} + 2312491545 x^{13} - 213869688 x^{12} - 490418469 x^{11} + 44511588 x^{10} + 65361426 x^{9} - 5507676 x^{8} - 5123475 x^{7} + 385695 x^{6} + 214380 x^{5} - 14391 x^{4} - 4176 x^{3} + 243 x^{2} + 27 x - 1 \)

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

Invariants

Degree:  $45$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[45, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(940\!\cdots\!689\)\(\medspace = 3^{110}\cdot 11^{36}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $99.86$
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:  $3, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $45$
This field is Galois and abelian over $\Q$.
Conductor:  \(297=3^{3}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{297}(256,·)$, $\chi_{297}(1,·)$, $\chi_{297}(130,·)$, $\chi_{297}(4,·)$, $\chi_{297}(133,·)$, $\chi_{297}(262,·)$, $\chi_{297}(136,·)$, $\chi_{297}(265,·)$, $\chi_{297}(268,·)$, $\chi_{297}(16,·)$, $\chi_{297}(148,·)$, $\chi_{297}(280,·)$, $\chi_{297}(25,·)$, $\chi_{297}(157,·)$, $\chi_{297}(31,·)$, $\chi_{297}(289,·)$, $\chi_{297}(34,·)$, $\chi_{297}(163,·)$, $\chi_{297}(37,·)$, $\chi_{297}(166,·)$, $\chi_{297}(295,·)$, $\chi_{297}(169,·)$, $\chi_{297}(49,·)$, $\chi_{297}(181,·)$, $\chi_{297}(58,·)$, $\chi_{297}(190,·)$, $\chi_{297}(64,·)$, $\chi_{297}(67,·)$, $\chi_{297}(196,·)$, $\chi_{297}(70,·)$, $\chi_{297}(199,·)$, $\chi_{297}(202,·)$, $\chi_{297}(82,·)$, $\chi_{297}(214,·)$, $\chi_{297}(91,·)$, $\chi_{297}(223,·)$, $\chi_{297}(97,·)$, $\chi_{297}(100,·)$, $\chi_{297}(229,·)$, $\chi_{297}(103,·)$, $\chi_{297}(232,·)$, $\chi_{297}(235,·)$, $\chi_{297}(115,·)$, $\chi_{297}(247,·)$, $\chi_{297}(124,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{109} a^{37} + \frac{41}{109} a^{36} - \frac{18}{109} a^{35} - \frac{51}{109} a^{34} + \frac{44}{109} a^{33} - \frac{43}{109} a^{32} + \frac{1}{109} a^{31} - \frac{12}{109} a^{30} - \frac{2}{109} a^{29} - \frac{33}{109} a^{28} + \frac{13}{109} a^{27} + \frac{49}{109} a^{26} + \frac{45}{109} a^{25} - \frac{53}{109} a^{24} + \frac{38}{109} a^{23} - \frac{38}{109} a^{22} - \frac{18}{109} a^{21} + \frac{6}{109} a^{20} - \frac{12}{109} a^{19} - \frac{18}{109} a^{18} - \frac{14}{109} a^{17} - \frac{40}{109} a^{16} + \frac{6}{109} a^{15} + \frac{16}{109} a^{14} + \frac{31}{109} a^{13} + \frac{23}{109} a^{12} - \frac{32}{109} a^{11} + \frac{12}{109} a^{10} - \frac{11}{109} a^{9} - \frac{21}{109} a^{8} - \frac{16}{109} a^{7} + \frac{37}{109} a^{6} - \frac{35}{109} a^{5} + \frac{26}{109} a^{4} - \frac{54}{109} a^{3} - \frac{32}{109} a^{2} + \frac{18}{109} a - \frac{5}{109}$, $\frac{1}{109} a^{38} + \frac{45}{109} a^{36} + \frac{33}{109} a^{35} - \frac{45}{109} a^{34} + \frac{6}{109} a^{33} + \frac{20}{109} a^{32} - \frac{53}{109} a^{31} + \frac{54}{109} a^{30} + \frac{49}{109} a^{29} - \frac{51}{109} a^{28} - \frac{48}{109} a^{27} - \frac{2}{109} a^{26} - \frac{45}{109} a^{25} + \frac{31}{109} a^{24} + \frac{39}{109} a^{23} + \frac{14}{109} a^{22} - \frac{19}{109} a^{21} - \frac{40}{109} a^{20} + \frac{38}{109} a^{19} - \frac{39}{109} a^{18} - \frac{11}{109} a^{17} + \frac{11}{109} a^{16} - \frac{12}{109} a^{15} + \frac{29}{109} a^{14} - \frac{49}{109} a^{13} + \frac{6}{109} a^{12} + \frac{16}{109} a^{11} + \frac{42}{109} a^{10} - \frac{6}{109} a^{9} - \frac{27}{109} a^{8} + \frac{39}{109} a^{7} - \frac{26}{109} a^{6} + \frac{44}{109} a^{5} - \frac{30}{109} a^{4} + \frac{2}{109} a^{3} + \frac{22}{109} a^{2} + \frac{20}{109} a - \frac{13}{109}$, $\frac{1}{109} a^{39} + \frac{41}{109} a^{36} + \frac{2}{109} a^{35} + \frac{12}{109} a^{34} + \frac{2}{109} a^{33} + \frac{29}{109} a^{32} + \frac{9}{109} a^{31} + \frac{44}{109} a^{30} + \frac{39}{109} a^{29} + \frac{20}{109} a^{28} - \frac{42}{109} a^{27} + \frac{39}{109} a^{26} - \frac{32}{109} a^{25} + \frac{26}{109} a^{24} + \frac{48}{109} a^{23} - \frac{53}{109} a^{22} + \frac{7}{109} a^{21} - \frac{14}{109} a^{20} - \frac{44}{109} a^{19} + \frac{36}{109} a^{18} - \frac{13}{109} a^{17} + \frac{44}{109} a^{16} - \frac{23}{109} a^{15} - \frac{6}{109} a^{14} + \frac{28}{109} a^{13} - \frac{38}{109} a^{12} - \frac{44}{109} a^{11} - \frac{1}{109} a^{10} + \frac{32}{109} a^{9} + \frac{3}{109} a^{8} + \frac{40}{109} a^{7} + \frac{14}{109} a^{6} + \frac{19}{109} a^{5} + \frac{31}{109} a^{4} + \frac{54}{109} a^{3} + \frac{43}{109} a^{2} + \frac{49}{109} a + \frac{7}{109}$, $\frac{1}{109} a^{40} - \frac{44}{109} a^{36} - \frac{13}{109} a^{35} + \frac{22}{109} a^{34} - \frac{31}{109} a^{33} + \frac{28}{109} a^{32} + \frac{3}{109} a^{31} - \frac{14}{109} a^{30} - \frac{7}{109} a^{29} + \frac{3}{109} a^{28} + \frac{51}{109} a^{27} + \frac{30}{109} a^{26} + \frac{34}{109} a^{25} + \frac{41}{109} a^{24} + \frac{24}{109} a^{23} + \frac{39}{109} a^{22} - \frac{39}{109} a^{21} + \frac{37}{109} a^{20} - \frac{17}{109} a^{19} - \frac{38}{109} a^{18} - \frac{36}{109} a^{17} - \frac{18}{109} a^{16} - \frac{34}{109} a^{15} + \frac{26}{109} a^{14} - \frac{1}{109} a^{13} - \frac{6}{109} a^{12} + \frac{3}{109} a^{11} - \frac{24}{109} a^{10} + \frac{18}{109} a^{9} + \frac{29}{109} a^{8} + \frac{16}{109} a^{7} + \frac{28}{109} a^{6} + \frac{49}{109} a^{5} - \frac{31}{109} a^{4} - \frac{32}{109} a^{3} + \frac{53}{109} a^{2} + \frac{32}{109} a - \frac{13}{109}$, $\frac{1}{109} a^{41} + \frac{47}{109} a^{36} - \frac{7}{109} a^{35} + \frac{14}{109} a^{34} + \frac{2}{109} a^{33} - \frac{36}{109} a^{32} + \frac{30}{109} a^{31} + \frac{10}{109} a^{30} + \frac{24}{109} a^{29} + \frac{16}{109} a^{28} - \frac{52}{109} a^{27} + \frac{10}{109} a^{26} - \frac{50}{109} a^{25} - \frac{19}{109} a^{24} - \frac{33}{109} a^{23} + \frac{33}{109} a^{22} + \frac{8}{109} a^{21} + \frac{29}{109} a^{20} - \frac{21}{109} a^{19} + \frac{44}{109} a^{18} + \frac{20}{109} a^{17} - \frac{50}{109} a^{16} - \frac{37}{109} a^{15} + \frac{49}{109} a^{14} + \frac{50}{109} a^{13} + \frac{34}{109} a^{12} - \frac{15}{109} a^{11} + \frac{1}{109} a^{10} - \frac{19}{109} a^{9} - \frac{36}{109} a^{8} - \frac{22}{109} a^{7} + \frac{42}{109} a^{6} - \frac{45}{109} a^{5} + \frac{22}{109} a^{4} - \frac{34}{109} a^{3} + \frac{41}{109} a^{2} + \frac{16}{109} a - \frac{2}{109}$, $\frac{1}{109} a^{42} + \frac{28}{109} a^{36} - \frac{12}{109} a^{35} + \frac{1}{109} a^{34} - \frac{33}{109} a^{33} - \frac{20}{109} a^{32} - \frac{37}{109} a^{31} + \frac{43}{109} a^{30} + \frac{1}{109} a^{29} - \frac{27}{109} a^{28} + \frac{53}{109} a^{27} + \frac{45}{109} a^{26} + \frac{46}{109} a^{25} - \frac{49}{109} a^{24} - \frac{9}{109} a^{23} + \frac{50}{109} a^{22} + \frac{3}{109} a^{21} + \frac{24}{109} a^{20} - \frac{46}{109} a^{19} - \frac{6}{109} a^{18} - \frac{46}{109} a^{17} - \frac{10}{109} a^{16} - \frac{15}{109} a^{15} - \frac{48}{109} a^{14} - \frac{6}{109} a^{13} - \frac{6}{109} a^{12} - \frac{21}{109} a^{11} - \frac{38}{109} a^{10} + \frac{45}{109} a^{9} - \frac{16}{109} a^{8} + \frac{31}{109} a^{7} - \frac{40}{109} a^{6} + \frac{32}{109} a^{5} + \frac{52}{109} a^{4} - \frac{37}{109} a^{3} - \frac{6}{109} a^{2} + \frac{24}{109} a + \frac{17}{109}$, $\frac{1}{109} a^{43} + \frac{39}{109} a^{36} - \frac{40}{109} a^{35} - \frac{22}{109} a^{34} - \frac{53}{109} a^{33} - \frac{32}{109} a^{32} + \frac{15}{109} a^{31} + \frac{10}{109} a^{30} + \frac{29}{109} a^{29} - \frac{4}{109} a^{28} + \frac{8}{109} a^{27} - \frac{18}{109} a^{26} - \frac{1}{109} a^{25} - \frac{51}{109} a^{24} - \frac{33}{109} a^{23} - \frac{23}{109} a^{22} - \frac{17}{109} a^{21} + \frac{4}{109} a^{20} + \frac{3}{109} a^{19} + \frac{22}{109} a^{18} - \frac{54}{109} a^{17} + \frac{15}{109} a^{16} + \frac{2}{109} a^{15} - \frac{18}{109} a^{14} - \frac{2}{109} a^{13} - \frac{11}{109} a^{12} - \frac{14}{109} a^{11} + \frac{36}{109} a^{10} - \frac{35}{109} a^{9} - \frac{35}{109} a^{8} - \frac{28}{109} a^{7} - \frac{23}{109} a^{6} + \frac{51}{109} a^{5} - \frac{2}{109} a^{4} - \frac{20}{109} a^{3} + \frac{48}{109} a^{2} - \frac{51}{109} a + \frac{31}{109}$, $\frac{1}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{44} - \frac{34510161837832718989181733471083659022135632983228315950760903590029267790162023416088557663371294422343346105610802225147300034}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{43} - \frac{24370919243576715419359656502705354520162078276842241169165209325939814971169429497784408426366549656402484795226187707455984480}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{42} - \frac{36068563211905298160372988389484446412948828221457965035268233633192947634414858442494308934244323337501860537700928089160910378}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{41} - \frac{30458597206403521097542463137391340828257358120749878483818432332303065273332049974876587101895810910243430691080086262106173297}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{40} - \frac{13318886083149367941933491379711234604022968858020251731378309500597703313114499414083521610644396024904936678101991813402480253}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{39} - \frac{51025884146732677203224671007844250398148858943862191018001407718514234958250089786403887976281256435456942088388039481747055128}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{38} + \frac{58131906378210992935479040011599766125106155885072436199939938442209672423413427824564769603683969110357827558967248205678784746}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{37} - \frac{2196843460026742621082867109906092737119651966955835868951382587117022605731357612537265610934567461186857840023149753505623720527}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{36} - \frac{6781034212498701177743629430423975285628795043119343930922841165604704947347368880106823567165270344981854864391131835743841694620}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{35} + \frac{4314997466157412444341625829438508268234954948145536720220051028957846856676285605604939009051810898156011036695208205309056811142}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{34} + \frac{4418240115838510707317218225475262793125532143634886527614207394309829721502371371031522102661148444778830658023097515174733642251}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{33} - \frac{2954266267178362132189959303505751217027031613816009557168669702687729588413720337934346309956358034533180035083016926280517658627}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{32} + \frac{7144289702209995337971591401379370095685265014472197964009949474278274119466025264295783863894003130885247636470960977516941471426}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{31} + \frac{824743914864376641944972362249792857388170018191552483271642653134218404203694293802731427751604495291491413305217375942460004309}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{30} - \frac{4608721201379916471221444907999099250378289663409880310133632324992680533304083888973028915164355443297556027152484579665615349621}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{29} + \frac{3012935543015905345910762462244755613579437175356559133768998643532062301673829816404890849604943502736245140991476115813991570290}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{28} - \frac{444445055035727449233652695805344960930838667783036474803110111587063764719638152535686422981046146978711821345935599362136326173}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{27} - \frac{2002995585116814015340451456910123350622332195837359085666233703137042944982795242935813391425286259065414791281977180769569181940}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{26} - \frac{2282671687400214008946694022812437750955157359889416379374554407907346838026604326840371723018433847615243395761045886511305894176}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{25} - \frac{844548158464790320873176392051510772870217020727528646909818419730767900358446346437212300864054348706688595922266784698142441610}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{24} + \frac{2830536658623795518316553437383942436291076942140319369333107589929893674099227609857154348770875491431109683060565090404028021280}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{23} + \frac{6070273227549659579099202126822002994991386743444137356661922374740676485516096580921353490743744951524181261796811140744981027765}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{22} - \frac{780129818225512250243320773901430736634103645498269105505070908577076253550821184449475171253406315463413776192696337507581936469}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{21} - \frac{5769319718836360430020745485911453600968354638456421058582095217592639586315592211265947216106333991697013222987667413845903419034}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{20} + \frac{3407873122330223419977468702939075922350293482712344977785726006569684572489072580772999226786059266068927824725724047987963568122}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{19} + \frac{5645075732206158283552151761882952387623280665203463332541568844654331079277244746176154862794130534788684618134337414219091524073}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{18} - \frac{2851450921928655866734511013209658093132815231888985694914194614590978825263211067491531637684876979887707934963226962362947600072}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{17} + \frac{2539674542538439777360874347521163363491076260880581373676379332078809222773382176598200258848973927094802391526965015721125777233}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{16} + \frac{4593860952878945907340501289211784525902482546280760736453029617252139197422641558905452533628443901339807349310801331870989673273}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{15} + \frac{7437117088709291227214160210755725317882136466582491704845850399909613435965516069146511338237679356488344188737133944072599460511}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{14} - \frac{71742679802423239440332443990094382137032409826825143696398802868756225150504232494645734635878807878327292533688826744139003536}{143847572809107276518510100891926783775595431731439617080978746847109812706317763873465477820503052168687467243768756583604727491} a^{13} - \frac{2589436761381103347122954011452293606538951991149565413278773632367044267751583171483409584668631845379639582620732178586165605270}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{12} - \frac{7636752388904707390624413386177016426473084713370966714227381261721554818171486570655766421216500964030724024476376494704083302661}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{11} + \frac{5682664447870714734464533681624753559812858761319130415810274848190520568861502071207843651746172381605472997983237386055921794941}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{10} - \frac{7308559495462490776968531751194117393359444333607021169584266758619689273778832708202343222523547784137320957820021083819424729696}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{9} - \frac{5480084812021900715416913490335650713374046027303106703401463007501811748847480624589011122528330215530468016775234529649563651662}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{8} + \frac{378248393676880108280831219565032772683061824285747253612092956760548740416599592398507353780130496234820724460369185874951944481}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{7} + \frac{497809567213734753427538667433588108805878570624098872356750300693149479752670477976173935843753678432108771566449908411790556382}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{6} + \frac{658455064916197276949017817707825263000718668020899244317264250386587350976561612149692183392080159996492069091391975500518202172}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{5} - \frac{4612405843654646865542377379545454612852993355688713294617822770146909674831054250928381914248026438172366685751736534556695768784}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{4} + \frac{5042522937326415632983237741545623237245303519482199457124445264564740172822559329264617139477238100106874210518818177594479799511}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{3} - \frac{6245823202182366715821668315104321394732738796330303786411278976514405958974870153571865402886619768067518378533288786454166486755}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a^{2} - \frac{602146972396659568697257308962571033714714567261784778520990652710079959105921372288937189519709040565856733461319565949319202817}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519} a + \frac{3213626405320569810596716738493040987057687888099682081135034763470831991462169752064575464361078363188155731317222967355397363308}{15679385436192693140517600997220019431539902058726918261826683406334969584988636262207737082434832686386933929570794467612915296519}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $44$
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:  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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 7652474177847196000000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{45}\cdot(2\pi)^{0}\cdot 7652474177847196000000000000000 \cdot 1}{2\sqrt{940750991812442660132286068374700929301371405411689329186316523736000865610069804102171689}}\approx 0.138798361578569$ (assuming GRH)

Galois group

$C_{45}$ (as 45T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 45
The 45 conjugacy class representatives for $C_{45}$
Character table for $C_{45}$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{27})^+\), 15.15.10943023107606534329121.1

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 $45$ R $45$ $45$ R $45$ $15^{3}$ $15^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{5}$ $45$ $45$ $15^{3}$ $45$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{5}$ $45$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{9}$ $45$

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
3Data not computed
11Data not computed