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Properties

Label 45.45.113...625.1
Degree $45$
Signature $[45, 0]$
Discriminant $1.132\times 10^{113}$
Root discriminant \(325.32\)
Ramified primes $5,37$
Class number not computed
Class group not computed
Galois group $C_{45}$ (as 45T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 5*x^44 - 160*x^43 + 750*x^42 + 11505*x^41 - 50155*x^40 - 494360*x^39 + 1986075*x^38 + 14230290*x^37 - 52164095*x^36 - 291406682*x^35 + 964545715*x^34 + 4397605380*x^33 - 13004815240*x^32 - 49959045085*x^31 + 130642800011*x^30 + 432522297350*x^29 - 990600139100*x^28 - 2869775308765*x^27 + 5708642080635*x^26 + 14596727897136*x^25 - 25057837740180*x^24 - 56657367198895*x^23 + 83639181826930*x^22 + 166254328426775*x^21 - 211229538265138*x^20 - 363413515659395*x^19 + 400345366164140*x^18 + 579295247175435*x^17 - 562802728668865*x^16 - 653353980492525*x^15 + 576820421117985*x^14 + 498607009071010*x^13 - 419400709761500*x^12 - 239088316288415*x^11 + 206781182240081*x^10 + 61592552138015*x^9 - 64047534431150*x^8 - 4385990367335*x^7 + 10818687121615*x^6 - 1100963095469*x^5 - 726031492295*x^4 + 153651014240*x^3 + 10678696865*x^2 - 4993044930*x + 335982349)
 
gp: K = bnfinit(y^45 - 5*y^44 - 160*y^43 + 750*y^42 + 11505*y^41 - 50155*y^40 - 494360*y^39 + 1986075*y^38 + 14230290*y^37 - 52164095*y^36 - 291406682*y^35 + 964545715*y^34 + 4397605380*y^33 - 13004815240*y^32 - 49959045085*y^31 + 130642800011*y^30 + 432522297350*y^29 - 990600139100*y^28 - 2869775308765*y^27 + 5708642080635*y^26 + 14596727897136*y^25 - 25057837740180*y^24 - 56657367198895*y^23 + 83639181826930*y^22 + 166254328426775*y^21 - 211229538265138*y^20 - 363413515659395*y^19 + 400345366164140*y^18 + 579295247175435*y^17 - 562802728668865*y^16 - 653353980492525*y^15 + 576820421117985*y^14 + 498607009071010*y^13 - 419400709761500*y^12 - 239088316288415*y^11 + 206781182240081*y^10 + 61592552138015*y^9 - 64047534431150*y^8 - 4385990367335*y^7 + 10818687121615*y^6 - 1100963095469*y^5 - 726031492295*y^4 + 153651014240*y^3 + 10678696865*y^2 - 4993044930*y + 335982349, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 5*x^44 - 160*x^43 + 750*x^42 + 11505*x^41 - 50155*x^40 - 494360*x^39 + 1986075*x^38 + 14230290*x^37 - 52164095*x^36 - 291406682*x^35 + 964545715*x^34 + 4397605380*x^33 - 13004815240*x^32 - 49959045085*x^31 + 130642800011*x^30 + 432522297350*x^29 - 990600139100*x^28 - 2869775308765*x^27 + 5708642080635*x^26 + 14596727897136*x^25 - 25057837740180*x^24 - 56657367198895*x^23 + 83639181826930*x^22 + 166254328426775*x^21 - 211229538265138*x^20 - 363413515659395*x^19 + 400345366164140*x^18 + 579295247175435*x^17 - 562802728668865*x^16 - 653353980492525*x^15 + 576820421117985*x^14 + 498607009071010*x^13 - 419400709761500*x^12 - 239088316288415*x^11 + 206781182240081*x^10 + 61592552138015*x^9 - 64047534431150*x^8 - 4385990367335*x^7 + 10818687121615*x^6 - 1100963095469*x^5 - 726031492295*x^4 + 153651014240*x^3 + 10678696865*x^2 - 4993044930*x + 335982349);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 5*x^44 - 160*x^43 + 750*x^42 + 11505*x^41 - 50155*x^40 - 494360*x^39 + 1986075*x^38 + 14230290*x^37 - 52164095*x^36 - 291406682*x^35 + 964545715*x^34 + 4397605380*x^33 - 13004815240*x^32 - 49959045085*x^31 + 130642800011*x^30 + 432522297350*x^29 - 990600139100*x^28 - 2869775308765*x^27 + 5708642080635*x^26 + 14596727897136*x^25 - 25057837740180*x^24 - 56657367198895*x^23 + 83639181826930*x^22 + 166254328426775*x^21 - 211229538265138*x^20 - 363413515659395*x^19 + 400345366164140*x^18 + 579295247175435*x^17 - 562802728668865*x^16 - 653353980492525*x^15 + 576820421117985*x^14 + 498607009071010*x^13 - 419400709761500*x^12 - 239088316288415*x^11 + 206781182240081*x^10 + 61592552138015*x^9 - 64047534431150*x^8 - 4385990367335*x^7 + 10818687121615*x^6 - 1100963095469*x^5 - 726031492295*x^4 + 153651014240*x^3 + 10678696865*x^2 - 4993044930*x + 335982349)
 

\( x^{45} - 5 x^{44} - 160 x^{43} + 750 x^{42} + 11505 x^{41} - 50155 x^{40} - 494360 x^{39} + \cdots + 335982349 \) Copy content Toggle raw display

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

Invariants

Degree:  $45$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[45, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(113\!\cdots\!625\) \(\medspace = 5^{72}\cdot 37^{40}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(325.32\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{8/5}37^{8/9}\approx 325.31875006938$
Ramified primes:   \(5\), \(37\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $45$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(925=5^{2}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{925}(256,·)$, $\chi_{925}(1,·)$, $\chi_{925}(386,·)$, $\chi_{925}(491,·)$, $\chi_{925}(641,·)$, $\chi_{925}(266,·)$, $\chi_{925}(396,·)$, $\chi_{925}(271,·)$, $\chi_{925}(16,·)$, $\chi_{925}(786,·)$, $\chi_{925}(921,·)$, $\chi_{925}(26,·)$, $\chi_{925}(416,·)$, $\chi_{925}(676,·)$, $\chi_{925}(551,·)$, $\chi_{925}(811,·)$, $\chi_{925}(556,·)$, $\chi_{925}(46,·)$, $\chi_{925}(306,·)$, $\chi_{925}(181,·)$, $\chi_{925}(441,·)$, $\chi_{925}(186,·)$, $\chi_{925}(571,·)$, $\chi_{925}(821,·)$, $\chi_{925}(451,·)$, $\chi_{925}(581,·)$, $\chi_{925}(71,·)$, $\chi_{925}(456,·)$, $\chi_{925}(201,·)$, $\chi_{925}(81,·)$, $\chi_{925}(211,·)$, $\chi_{925}(86,·)$, $\chi_{925}(601,·)$, $\chi_{925}(861,·)$, $\chi_{925}(736,·)$, $\chi_{925}(741,·)$, $\chi_{925}(231,·)$, $\chi_{925}(826,·)$, $\chi_{925}(366,·)$, $\chi_{925}(626,·)$, $\chi_{925}(371,·)$, $\chi_{925}(756,·)$, $\chi_{925}(121,·)$, $\chi_{925}(636,·)$, $\chi_{925}(766,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{43}a^{31}+\frac{6}{43}a^{30}+\frac{18}{43}a^{28}-\frac{2}{43}a^{27}-\frac{8}{43}a^{26}-\frac{13}{43}a^{25}+\frac{11}{43}a^{24}-\frac{18}{43}a^{23}-\frac{17}{43}a^{22}+\frac{16}{43}a^{21}+\frac{5}{43}a^{20}+\frac{8}{43}a^{19}+\frac{13}{43}a^{18}-\frac{12}{43}a^{17}-\frac{8}{43}a^{16}-\frac{11}{43}a^{15}+\frac{10}{43}a^{14}-\frac{8}{43}a^{13}-\frac{18}{43}a^{12}+\frac{5}{43}a^{11}+\frac{8}{43}a^{10}+\frac{10}{43}a^{9}-\frac{21}{43}a^{8}-\frac{21}{43}a^{7}-\frac{10}{43}a^{6}+\frac{11}{43}a^{4}+\frac{9}{43}a^{3}-\frac{7}{43}a$, $\frac{1}{43}a^{32}+\frac{7}{43}a^{30}+\frac{18}{43}a^{29}+\frac{19}{43}a^{28}+\frac{4}{43}a^{27}-\frac{8}{43}a^{26}+\frac{3}{43}a^{25}+\frac{2}{43}a^{24}+\frac{5}{43}a^{23}-\frac{11}{43}a^{22}-\frac{5}{43}a^{21}+\frac{21}{43}a^{20}+\frac{8}{43}a^{19}-\frac{4}{43}a^{18}+\frac{21}{43}a^{17}-\frac{6}{43}a^{16}-\frac{10}{43}a^{15}+\frac{18}{43}a^{14}-\frac{13}{43}a^{13}-\frac{16}{43}a^{12}+\frac{21}{43}a^{11}+\frac{5}{43}a^{10}+\frac{5}{43}a^{9}+\frac{19}{43}a^{8}-\frac{13}{43}a^{7}+\frac{17}{43}a^{6}+\frac{11}{43}a^{5}-\frac{14}{43}a^{4}-\frac{11}{43}a^{3}-\frac{7}{43}a^{2}-\frac{1}{43}a$, $\frac{1}{43}a^{33}+\frac{19}{43}a^{30}+\frac{19}{43}a^{29}+\frac{7}{43}a^{28}+\frac{6}{43}a^{27}+\frac{16}{43}a^{26}+\frac{7}{43}a^{25}+\frac{14}{43}a^{24}-\frac{14}{43}a^{23}-\frac{15}{43}a^{22}-\frac{5}{43}a^{21}+\frac{16}{43}a^{20}-\frac{17}{43}a^{19}+\frac{16}{43}a^{18}-\frac{8}{43}a^{17}+\frac{3}{43}a^{16}+\frac{9}{43}a^{15}+\frac{3}{43}a^{14}-\frac{3}{43}a^{13}+\frac{18}{43}a^{12}+\frac{13}{43}a^{11}-\frac{8}{43}a^{10}-\frac{8}{43}a^{9}+\frac{5}{43}a^{8}-\frac{8}{43}a^{7}-\frac{5}{43}a^{6}-\frac{14}{43}a^{5}-\frac{2}{43}a^{4}+\frac{16}{43}a^{3}-\frac{1}{43}a^{2}+\frac{6}{43}a$, $\frac{1}{43}a^{34}-\frac{9}{43}a^{30}+\frac{7}{43}a^{29}+\frac{8}{43}a^{28}+\frac{11}{43}a^{27}-\frac{13}{43}a^{26}+\frac{3}{43}a^{25}-\frac{8}{43}a^{24}-\frac{17}{43}a^{23}+\frac{17}{43}a^{22}+\frac{13}{43}a^{21}+\frac{17}{43}a^{20}-\frac{7}{43}a^{19}+\frac{3}{43}a^{18}+\frac{16}{43}a^{17}-\frac{11}{43}a^{16}-\frac{3}{43}a^{15}-\frac{21}{43}a^{14}-\frac{2}{43}a^{13}+\frac{11}{43}a^{12}-\frac{17}{43}a^{11}+\frac{12}{43}a^{10}-\frac{13}{43}a^{9}+\frac{4}{43}a^{8}+\frac{7}{43}a^{7}+\frac{4}{43}a^{6}-\frac{2}{43}a^{5}-\frac{21}{43}a^{4}+\frac{6}{43}a^{2}+\frac{4}{43}a$, $\frac{1}{43}a^{35}+\frac{18}{43}a^{30}+\frac{8}{43}a^{29}+\frac{1}{43}a^{28}+\frac{12}{43}a^{27}+\frac{17}{43}a^{26}+\frac{4}{43}a^{25}-\frac{4}{43}a^{24}-\frac{16}{43}a^{23}-\frac{11}{43}a^{22}-\frac{11}{43}a^{21}-\frac{5}{43}a^{20}-\frac{11}{43}a^{19}+\frac{4}{43}a^{18}+\frac{10}{43}a^{17}+\frac{11}{43}a^{16}+\frac{9}{43}a^{15}+\frac{2}{43}a^{14}-\frac{18}{43}a^{13}-\frac{7}{43}a^{12}+\frac{14}{43}a^{11}+\frac{16}{43}a^{10}+\frac{8}{43}a^{9}-\frac{10}{43}a^{8}-\frac{13}{43}a^{7}-\frac{6}{43}a^{6}-\frac{21}{43}a^{5}+\frac{13}{43}a^{4}+\frac{1}{43}a^{3}+\frac{4}{43}a^{2}-\frac{20}{43}a$, $\frac{1}{301}a^{36}-\frac{3}{301}a^{35}-\frac{2}{301}a^{34}-\frac{2}{301}a^{33}+\frac{1}{301}a^{32}-\frac{2}{301}a^{31}-\frac{93}{301}a^{30}+\frac{72}{301}a^{29}-\frac{21}{43}a^{28}-\frac{138}{301}a^{27}+\frac{13}{301}a^{26}+\frac{141}{301}a^{25}-\frac{62}{301}a^{24}-\frac{9}{301}a^{23}+\frac{132}{301}a^{22}-\frac{14}{43}a^{21}-\frac{14}{43}a^{20}+\frac{15}{43}a^{19}+\frac{40}{301}a^{18}+\frac{97}{301}a^{17}-\frac{69}{301}a^{16}+\frac{87}{301}a^{15}+\frac{88}{301}a^{14}-\frac{54}{301}a^{13}-\frac{23}{301}a^{12}-\frac{20}{43}a^{11}-\frac{74}{301}a^{10}-\frac{144}{301}a^{9}-\frac{78}{301}a^{8}+\frac{12}{301}a^{7}+\frac{44}{301}a^{6}+\frac{17}{43}a^{5}-\frac{11}{301}a^{4}-\frac{136}{301}a^{3}-\frac{6}{301}a^{2}-\frac{36}{301}a+\frac{1}{7}$, $\frac{1}{301}a^{37}+\frac{3}{301}a^{35}-\frac{1}{301}a^{34}+\frac{2}{301}a^{33}+\frac{1}{301}a^{32}-\frac{1}{301}a^{31}+\frac{101}{301}a^{30}+\frac{62}{301}a^{29}+\frac{100}{301}a^{28}-\frac{9}{301}a^{27}-\frac{44}{301}a^{26}+\frac{116}{301}a^{25}-\frac{34}{301}a^{24}+\frac{1}{43}a^{23}-\frac{3}{301}a^{22}-\frac{18}{43}a^{21}-\frac{20}{43}a^{20}-\frac{2}{7}a^{19}-\frac{18}{43}a^{18}+\frac{145}{301}a^{17}+\frac{97}{301}a^{16}+\frac{41}{301}a^{15}-\frac{16}{43}a^{14}-\frac{52}{301}a^{13}-\frac{62}{301}a^{12}-\frac{137}{301}a^{11}+\frac{68}{301}a^{10}+\frac{134}{301}a^{9}+\frac{51}{301}a^{8}-\frac{60}{301}a^{7}+\frac{83}{301}a^{6}-\frac{60}{301}a^{5}+\frac{27}{301}a^{4}-\frac{8}{301}a^{3}+\frac{37}{301}a^{2}-\frac{58}{301}a+\frac{3}{7}$, $\frac{1}{301}a^{38}+\frac{1}{301}a^{35}+\frac{1}{301}a^{34}+\frac{3}{301}a^{32}+\frac{2}{301}a^{31}-\frac{135}{301}a^{30}+\frac{73}{301}a^{29}+\frac{68}{301}a^{28}+\frac{104}{301}a^{27}+\frac{17}{43}a^{26}-\frac{72}{301}a^{25}-\frac{59}{301}a^{24}-\frac{130}{301}a^{23}+\frac{45}{301}a^{22}-\frac{5}{43}a^{21}-\frac{65}{301}a^{20}-\frac{11}{43}a^{19}-\frac{24}{301}a^{18}-\frac{117}{301}a^{17}+\frac{122}{301}a^{16}+\frac{5}{301}a^{15}+\frac{76}{301}a^{14}+\frac{107}{301}a^{13}+\frac{51}{301}a^{12}+\frac{40}{301}a^{11}+\frac{13}{301}a^{10}-\frac{20}{43}a^{9}+\frac{111}{301}a^{8}-\frac{149}{301}a^{7}+\frac{123}{301}a^{6}+\frac{6}{301}a^{5}+\frac{46}{301}a^{4}-\frac{94}{301}a^{3}+\frac{149}{301}a^{2}+\frac{132}{301}a-\frac{3}{7}$, $\frac{1}{301}a^{39}-\frac{3}{301}a^{35}+\frac{2}{301}a^{34}-\frac{2}{301}a^{33}+\frac{1}{301}a^{32}+\frac{103}{301}a^{30}+\frac{108}{301}a^{29}-\frac{120}{301}a^{28}-\frac{135}{301}a^{27}+\frac{125}{301}a^{26}+\frac{101}{301}a^{25}+\frac{121}{301}a^{24}-\frac{23}{301}a^{23}-\frac{139}{301}a^{22}-\frac{135}{301}a^{21}+\frac{1}{43}a^{20}-\frac{73}{301}a^{19}-\frac{73}{301}a^{18}-\frac{80}{301}a^{17}+\frac{116}{301}a^{16}-\frac{95}{301}a^{15}+\frac{110}{301}a^{14}+\frac{13}{43}a^{13}+\frac{27}{301}a^{11}+\frac{39}{301}a^{10}+\frac{80}{301}a^{9}-\frac{120}{301}a^{8}-\frac{127}{301}a^{7}-\frac{87}{301}a^{6}-\frac{3}{7}a^{5}+\frac{99}{301}a^{4}-\frac{142}{301}a^{3}+\frac{117}{301}a^{2}-\frac{23}{301}a-\frac{1}{7}$, $\frac{1}{9331}a^{40}-\frac{1}{1333}a^{39}+\frac{5}{9331}a^{38}-\frac{8}{9331}a^{37}-\frac{4}{9331}a^{36}+\frac{10}{1333}a^{35}+\frac{48}{9331}a^{34}+\frac{78}{9331}a^{33}-\frac{57}{9331}a^{32}+\frac{102}{9331}a^{31}+\frac{839}{9331}a^{30}-\frac{2787}{9331}a^{29}+\frac{121}{1333}a^{28}-\frac{4451}{9331}a^{27}+\frac{2743}{9331}a^{26}-\frac{2778}{9331}a^{25}-\frac{138}{9331}a^{24}+\frac{4029}{9331}a^{23}-\frac{2097}{9331}a^{22}-\frac{60}{1333}a^{21}-\frac{265}{9331}a^{20}+\frac{3695}{9331}a^{19}-\frac{1955}{9331}a^{18}-\frac{1292}{9331}a^{17}-\frac{1620}{9331}a^{16}+\frac{26}{1333}a^{15}+\frac{4338}{9331}a^{14}+\frac{3112}{9331}a^{13}-\frac{1859}{9331}a^{12}-\frac{751}{9331}a^{11}-\frac{66}{9331}a^{10}-\frac{4450}{9331}a^{9}+\frac{296}{1333}a^{8}-\frac{10}{31}a^{7}+\frac{3894}{9331}a^{6}+\frac{98}{1333}a^{5}-\frac{621}{9331}a^{4}+\frac{4068}{9331}a^{3}+\frac{2280}{9331}a^{2}+\frac{2384}{9331}a-\frac{47}{217}$, $\frac{1}{557517919}a^{41}-\frac{17299}{557517919}a^{40}+\frac{17447}{557517919}a^{39}-\frac{213816}{557517919}a^{38}-\frac{117834}{79645417}a^{37}-\frac{73428}{79645417}a^{36}+\frac{1745768}{557517919}a^{35}+\frac{6119766}{557517919}a^{34}+\frac{39845}{79645417}a^{33}-\frac{749942}{79645417}a^{32}+\frac{2104491}{557517919}a^{31}+\frac{22070027}{79645417}a^{30}+\frac{72362956}{557517919}a^{29}+\frac{249411317}{557517919}a^{28}+\frac{2970676}{17984449}a^{27}-\frac{89935641}{557517919}a^{26}+\frac{184512771}{557517919}a^{25}-\frac{174660967}{557517919}a^{24}-\frac{214794500}{557517919}a^{23}-\frac{72231842}{557517919}a^{22}-\frac{175580988}{557517919}a^{21}+\frac{141503031}{557517919}a^{20}-\frac{190038336}{557517919}a^{19}-\frac{271776506}{557517919}a^{18}+\frac{22674395}{79645417}a^{17}-\frac{155214727}{557517919}a^{16}-\frac{77148951}{557517919}a^{15}+\frac{8175458}{17984449}a^{14}-\frac{24085537}{79645417}a^{13}-\frac{31372745}{557517919}a^{12}+\frac{2594189}{557517919}a^{11}-\frac{244145191}{557517919}a^{10}+\frac{250181053}{557517919}a^{9}-\frac{66713552}{557517919}a^{8}-\frac{176665651}{557517919}a^{7}-\frac{257231279}{557517919}a^{6}+\frac{206864263}{557517919}a^{5}-\frac{277065444}{557517919}a^{4}-\frac{90561822}{557517919}a^{3}+\frac{30555731}{79645417}a^{2}+\frac{140694189}{557517919}a-\frac{237866}{12965533}$, $\frac{1}{23973270517}a^{42}+\frac{3}{23973270517}a^{41}-\frac{365604}{23973270517}a^{40}-\frac{3424216}{23973270517}a^{39}+\frac{33684736}{23973270517}a^{38}+\frac{694819}{3424752931}a^{37}-\frac{29560842}{23973270517}a^{36}-\frac{12198}{79645417}a^{35}+\frac{86796449}{23973270517}a^{34}+\frac{27353458}{23973270517}a^{33}-\frac{1079511}{23973270517}a^{32}-\frac{86272085}{23973270517}a^{31}+\frac{369100389}{3424752931}a^{30}+\frac{2712923023}{23973270517}a^{29}+\frac{10547590997}{23973270517}a^{28}-\frac{5269388721}{23973270517}a^{27}-\frac{10909394025}{23973270517}a^{26}+\frac{1752590471}{23973270517}a^{25}+\frac{6513694368}{23973270517}a^{24}+\frac{1102656677}{3424752931}a^{23}-\frac{11766485036}{23973270517}a^{22}-\frac{10904433626}{23973270517}a^{21}+\frac{5582394339}{23973270517}a^{20}+\frac{11524813152}{23973270517}a^{19}-\frac{4877618271}{23973270517}a^{18}+\frac{7523578588}{23973270517}a^{17}-\frac{6467771078}{23973270517}a^{16}-\frac{11577591636}{23973270517}a^{15}-\frac{525888570}{3424752931}a^{14}-\frac{5104238563}{23973270517}a^{13}-\frac{165174602}{23973270517}a^{12}-\frac{1310650543}{23973270517}a^{11}-\frac{439667926}{23973270517}a^{10}+\frac{971786269}{23973270517}a^{9}+\frac{487446757}{3424752931}a^{8}+\frac{87104133}{773331307}a^{7}+\frac{4036006381}{23973270517}a^{6}+\frac{68842678}{557517919}a^{5}-\frac{8091912070}{23973270517}a^{4}-\frac{6176114397}{23973270517}a^{3}-\frac{369413958}{3424752931}a^{2}+\frac{3042853943}{23973270517}a+\frac{18236782}{557517919}$, $\frac{1}{10620158839031}a^{43}-\frac{103}{10620158839031}a^{42}-\frac{5367}{10620158839031}a^{41}-\frac{493133183}{10620158839031}a^{40}+\frac{13079441733}{10620158839031}a^{39}+\frac{297599034}{246980438117}a^{38}+\frac{3878149109}{10620158839031}a^{37}-\frac{12207607385}{10620158839031}a^{36}+\frac{101985297475}{10620158839031}a^{35}-\frac{114566897370}{10620158839031}a^{34}-\frac{117949338088}{10620158839031}a^{33}-\frac{40713496604}{10620158839031}a^{32}-\frac{2640537787}{246980438117}a^{31}+\frac{1901553315691}{10620158839031}a^{30}-\frac{1787631922672}{10620158839031}a^{29}+\frac{3657117302395}{10620158839031}a^{28}-\frac{1162596769134}{10620158839031}a^{27}-\frac{407338722188}{10620158839031}a^{26}+\frac{5081470246929}{10620158839031}a^{25}+\frac{329387606966}{1517165548433}a^{24}+\frac{3534877148426}{10620158839031}a^{23}+\frac{67750122158}{246980438117}a^{22}-\frac{84447560136}{10620158839031}a^{21}+\frac{17921977029}{1517165548433}a^{20}-\frac{52890886502}{1517165548433}a^{19}-\frac{2467509813371}{10620158839031}a^{18}-\frac{2928423345931}{10620158839031}a^{17}-\frac{598094840893}{1517165548433}a^{16}-\frac{3399331526946}{10620158839031}a^{15}-\frac{164566001167}{10620158839031}a^{14}-\frac{5149611326158}{10620158839031}a^{13}-\frac{3783988985589}{10620158839031}a^{12}+\frac{3018400514238}{10620158839031}a^{11}-\frac{4462711038845}{10620158839031}a^{10}+\frac{3863059385616}{10620158839031}a^{9}-\frac{3716899939131}{10620158839031}a^{8}-\frac{3873389820277}{10620158839031}a^{7}+\frac{69441646076}{10620158839031}a^{6}+\frac{1202675071847}{10620158839031}a^{5}-\frac{860541380484}{10620158839031}a^{4}-\frac{2522572188952}{10620158839031}a^{3}-\frac{198290020541}{1517165548433}a^{2}+\frac{11402203961}{342585769001}a-\frac{39737487453}{246980438117}$, $\frac{1}{20\!\cdots\!19}a^{44}+\frac{13\!\cdots\!29}{29\!\cdots\!17}a^{43}+\frac{13\!\cdots\!77}{20\!\cdots\!19}a^{42}-\frac{11\!\cdots\!50}{20\!\cdots\!19}a^{41}+\frac{31\!\cdots\!35}{29\!\cdots\!17}a^{40}+\frac{92\!\cdots\!54}{20\!\cdots\!19}a^{39}-\frac{14\!\cdots\!29}{20\!\cdots\!19}a^{38}-\frac{10\!\cdots\!52}{20\!\cdots\!19}a^{37}-\frac{49\!\cdots\!12}{20\!\cdots\!19}a^{36}-\frac{20\!\cdots\!57}{20\!\cdots\!19}a^{35}+\frac{14\!\cdots\!91}{20\!\cdots\!19}a^{34}-\frac{41\!\cdots\!71}{20\!\cdots\!19}a^{33}+\frac{16\!\cdots\!36}{20\!\cdots\!19}a^{32}-\frac{19\!\cdots\!15}{20\!\cdots\!19}a^{31}-\frac{84\!\cdots\!74}{20\!\cdots\!19}a^{30}-\frac{67\!\cdots\!40}{20\!\cdots\!19}a^{29}-\frac{37\!\cdots\!06}{20\!\cdots\!19}a^{28}-\frac{40\!\cdots\!98}{20\!\cdots\!19}a^{27}-\frac{36\!\cdots\!11}{20\!\cdots\!19}a^{26}+\frac{35\!\cdots\!55}{20\!\cdots\!19}a^{25}-\frac{81\!\cdots\!24}{20\!\cdots\!19}a^{24}-\frac{18\!\cdots\!59}{20\!\cdots\!19}a^{23}-\frac{39\!\cdots\!78}{13\!\cdots\!31}a^{22}+\frac{97\!\cdots\!00}{20\!\cdots\!19}a^{21}-\frac{73\!\cdots\!68}{20\!\cdots\!19}a^{20}-\frac{35\!\cdots\!45}{29\!\cdots\!17}a^{19}+\frac{12\!\cdots\!25}{48\!\cdots\!33}a^{18}+\frac{64\!\cdots\!61}{20\!\cdots\!19}a^{17}+\frac{46\!\cdots\!61}{20\!\cdots\!19}a^{16}+\frac{57\!\cdots\!14}{20\!\cdots\!19}a^{15}-\frac{10\!\cdots\!25}{20\!\cdots\!19}a^{14}+\frac{30\!\cdots\!69}{20\!\cdots\!19}a^{13}+\frac{20\!\cdots\!64}{20\!\cdots\!19}a^{12}-\frac{77\!\cdots\!84}{20\!\cdots\!19}a^{11}+\frac{56\!\cdots\!21}{20\!\cdots\!19}a^{10}-\frac{29\!\cdots\!07}{29\!\cdots\!17}a^{9}-\frac{14\!\cdots\!04}{67\!\cdots\!49}a^{8}+\frac{10\!\cdots\!71}{20\!\cdots\!19}a^{7}-\frac{83\!\cdots\!44}{20\!\cdots\!19}a^{6}+\frac{91\!\cdots\!43}{20\!\cdots\!19}a^{5}+\frac{33\!\cdots\!52}{20\!\cdots\!19}a^{4}+\frac{92\!\cdots\!57}{20\!\cdots\!19}a^{3}-\frac{49\!\cdots\!75}{20\!\cdots\!19}a^{2}-\frac{94\!\cdots\!07}{20\!\cdots\!19}a+\frac{18\!\cdots\!24}{69\!\cdots\!19}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $43$

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $44$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^45 - 5*x^44 - 160*x^43 + 750*x^42 + 11505*x^41 - 50155*x^40 - 494360*x^39 + 1986075*x^38 + 14230290*x^37 - 52164095*x^36 - 291406682*x^35 + 964545715*x^34 + 4397605380*x^33 - 13004815240*x^32 - 49959045085*x^31 + 130642800011*x^30 + 432522297350*x^29 - 990600139100*x^28 - 2869775308765*x^27 + 5708642080635*x^26 + 14596727897136*x^25 - 25057837740180*x^24 - 56657367198895*x^23 + 83639181826930*x^22 + 166254328426775*x^21 - 211229538265138*x^20 - 363413515659395*x^19 + 400345366164140*x^18 + 579295247175435*x^17 - 562802728668865*x^16 - 653353980492525*x^15 + 576820421117985*x^14 + 498607009071010*x^13 - 419400709761500*x^12 - 239088316288415*x^11 + 206781182240081*x^10 + 61592552138015*x^9 - 64047534431150*x^8 - 4385990367335*x^7 + 10818687121615*x^6 - 1100963095469*x^5 - 726031492295*x^4 + 153651014240*x^3 + 10678696865*x^2 - 4993044930*x + 335982349)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^45 - 5*x^44 - 160*x^43 + 750*x^42 + 11505*x^41 - 50155*x^40 - 494360*x^39 + 1986075*x^38 + 14230290*x^37 - 52164095*x^36 - 291406682*x^35 + 964545715*x^34 + 4397605380*x^33 - 13004815240*x^32 - 49959045085*x^31 + 130642800011*x^30 + 432522297350*x^29 - 990600139100*x^28 - 2869775308765*x^27 + 5708642080635*x^26 + 14596727897136*x^25 - 25057837740180*x^24 - 56657367198895*x^23 + 83639181826930*x^22 + 166254328426775*x^21 - 211229538265138*x^20 - 363413515659395*x^19 + 400345366164140*x^18 + 579295247175435*x^17 - 562802728668865*x^16 - 653353980492525*x^15 + 576820421117985*x^14 + 498607009071010*x^13 - 419400709761500*x^12 - 239088316288415*x^11 + 206781182240081*x^10 + 61592552138015*x^9 - 64047534431150*x^8 - 4385990367335*x^7 + 10818687121615*x^6 - 1100963095469*x^5 - 726031492295*x^4 + 153651014240*x^3 + 10678696865*x^2 - 4993044930*x + 335982349, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^45 - 5*x^44 - 160*x^43 + 750*x^42 + 11505*x^41 - 50155*x^40 - 494360*x^39 + 1986075*x^38 + 14230290*x^37 - 52164095*x^36 - 291406682*x^35 + 964545715*x^34 + 4397605380*x^33 - 13004815240*x^32 - 49959045085*x^31 + 130642800011*x^30 + 432522297350*x^29 - 990600139100*x^28 - 2869775308765*x^27 + 5708642080635*x^26 + 14596727897136*x^25 - 25057837740180*x^24 - 56657367198895*x^23 + 83639181826930*x^22 + 166254328426775*x^21 - 211229538265138*x^20 - 363413515659395*x^19 + 400345366164140*x^18 + 579295247175435*x^17 - 562802728668865*x^16 - 653353980492525*x^15 + 576820421117985*x^14 + 498607009071010*x^13 - 419400709761500*x^12 - 239088316288415*x^11 + 206781182240081*x^10 + 61592552138015*x^9 - 64047534431150*x^8 - 4385990367335*x^7 + 10818687121615*x^6 - 1100963095469*x^5 - 726031492295*x^4 + 153651014240*x^3 + 10678696865*x^2 - 4993044930*x + 335982349);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 5*x^44 - 160*x^43 + 750*x^42 + 11505*x^41 - 50155*x^40 - 494360*x^39 + 1986075*x^38 + 14230290*x^37 - 52164095*x^36 - 291406682*x^35 + 964545715*x^34 + 4397605380*x^33 - 13004815240*x^32 - 49959045085*x^31 + 130642800011*x^30 + 432522297350*x^29 - 990600139100*x^28 - 2869775308765*x^27 + 5708642080635*x^26 + 14596727897136*x^25 - 25057837740180*x^24 - 56657367198895*x^23 + 83639181826930*x^22 + 166254328426775*x^21 - 211229538265138*x^20 - 363413515659395*x^19 + 400345366164140*x^18 + 579295247175435*x^17 - 562802728668865*x^16 - 653353980492525*x^15 + 576820421117985*x^14 + 498607009071010*x^13 - 419400709761500*x^12 - 239088316288415*x^11 + 206781182240081*x^10 + 61592552138015*x^9 - 64047534431150*x^8 - 4385990367335*x^7 + 10818687121615*x^6 - 1100963095469*x^5 - 726031492295*x^4 + 153651014240*x^3 + 10678696865*x^2 - 4993044930*x + 335982349);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

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

Intermediate fields

3.3.1369.1, 5.5.390625.1, 9.9.3512479453921.1, 15.15.286613963390460550785064697265625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $45$ $45$ R ${\href{/padicField/7.9.0.1}{9} }^{5}$ $15^{3}$ $45$ $45$ $45$ $15^{3}$ $15^{3}$ ${\href{/padicField/31.5.0.1}{5} }^{9}$ R $45$ ${\href{/padicField/43.1.0.1}{1} }^{45}$ $15^{3}$ $45$ $45$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display Deg $45$$5$$9$$72$
\(37\) Copy content Toggle raw display Deg $45$$9$$5$$40$