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Properties

Label 45.45.202...625.1
Degree $45$
Signature $[45, 0]$
Discriminant $2.023\times 10^{104}$
Root discriminant \(207.93\)
Ramified primes $3,5,7$
Class number not computed
Class group not computed
Galois group $C_3\times C_{15}$ (as 45T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757)
 
gp: K = bnfinit(y^45 - 165*y^43 - 25*y^42 + 12060*y^41 + 3486*y^40 - 519330*y^39 - 214140*y^38 + 14776560*y^37 + 7718745*y^36 - 295077669*y^35 - 182852610*y^34 + 4284179615*y^33 + 3016157820*y^32 - 46197393465*y^31 - 35772719505*y^30 + 374607222195*y^29 + 310484223975*y^28 - 2298730688540*y^27 - 1988244775980*y^26 + 10695933805596*y^25 + 9407317026635*y^24 - 37696094037390*y^23 - 32764647222885*y^22 + 100289481110340*y^21 + 83332538318277*y^20 - 200279250507195*y^19 - 152889355168310*y^18 + 297466429398705*y^17 + 198745289480940*y^16 - 323624520119411*y^15 - 178210655500665*y^14 + 251547979741875*y^13 + 105785393431525*y^12 - 134218212107550*y^11 - 38927292179565*y^10 + 46104688292390*y^9 + 7956837983055*y^8 - 9173261413650*y^7 - 765305596255*y^6 + 891053703312*y^5 + 40709257785*y^4 - 38148076985*y^3 - 1973427960*y^2 + 606052050*y + 47691757, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757)
 

\( x^{45} - 165 x^{43} - 25 x^{42} + 12060 x^{41} + 3486 x^{40} - 519330 x^{39} - 214140 x^{38} + \cdots + 47691757 \) 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:   \(202\!\cdots\!625\) \(\medspace = 3^{60}\cdot 5^{72}\cdot 7^{30}\) 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:  \(207.93\)
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:  $3^{4/3}5^{8/5}7^{2/3}\approx 207.92771130440346$
Ramified primes:   \(3\), \(5\), \(7\) 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:  \(1575=3^{2}\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1575}(256,·)$, $\chi_{1575}(1,·)$, $\chi_{1575}(1411,·)$, $\chi_{1575}(1156,·)$, $\chi_{1575}(646,·)$, $\chi_{1575}(781,·)$, $\chi_{1575}(526,·)$, $\chi_{1575}(16,·)$, $\chi_{1575}(1171,·)$, $\chi_{1575}(151,·)$, $\chi_{1575}(1306,·)$, $\chi_{1575}(1051,·)$, $\chi_{1575}(541,·)$, $\chi_{1575}(676,·)$, $\chi_{1575}(421,·)$, $\chi_{1575}(1066,·)$, $\chi_{1575}(46,·)$, $\chi_{1575}(1201,·)$, $\chi_{1575}(946,·)$, $\chi_{1575}(436,·)$, $\chi_{1575}(571,·)$, $\chi_{1575}(316,·)$, $\chi_{1575}(1471,·)$, $\chi_{1575}(961,·)$, $\chi_{1575}(1096,·)$, $\chi_{1575}(841,·)$, $\chi_{1575}(331,·)$, $\chi_{1575}(1486,·)$, $\chi_{1575}(466,·)$, $\chi_{1575}(211,·)$, $\chi_{1575}(1366,·)$, $\chi_{1575}(856,·)$, $\chi_{1575}(991,·)$, $\chi_{1575}(736,·)$, $\chi_{1575}(226,·)$, $\chi_{1575}(1381,·)$, $\chi_{1575}(361,·)$, $\chi_{1575}(106,·)$, $\chi_{1575}(1516,·)$, $\chi_{1575}(1261,·)$, $\chi_{1575}(751,·)$, $\chi_{1575}(886,·)$, $\chi_{1575}(631,·)$, $\chi_{1575}(121,·)$, $\chi_{1575}(1276,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $\frac{1}{2}a^{35}-\frac{1}{2}a^{32}-\frac{1}{2}a^{30}-\frac{1}{2}a^{29}-\frac{1}{2}a^{28}-\frac{1}{2}a^{26}-\frac{1}{2}a^{24}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{14}a^{36}+\frac{1}{14}a^{35}-\frac{2}{7}a^{34}+\frac{1}{14}a^{33}-\frac{1}{2}a^{32}-\frac{1}{2}a^{31}-\frac{2}{7}a^{30}+\frac{1}{7}a^{29}-\frac{1}{2}a^{28}+\frac{5}{14}a^{27}+\frac{5}{14}a^{26}-\frac{1}{14}a^{25}-\frac{3}{14}a^{24}-\frac{2}{7}a^{23}-\frac{2}{7}a^{22}+\frac{3}{7}a^{21}+\frac{2}{7}a^{20}-\frac{3}{7}a^{19}+\frac{3}{7}a^{18}-\frac{1}{7}a^{17}+\frac{3}{7}a^{16}-\frac{2}{7}a^{14}+\frac{3}{14}a^{13}-\frac{1}{2}a^{12}-\frac{3}{7}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{2}{7}a^{8}+\frac{5}{14}a^{7}-\frac{3}{7}a^{6}-\frac{1}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}-\frac{1}{2}a^{2}+\frac{3}{14}a+\frac{1}{14}$, $\frac{1}{14}a^{37}+\frac{1}{7}a^{35}+\frac{5}{14}a^{34}+\frac{3}{7}a^{33}-\frac{1}{2}a^{32}+\frac{3}{14}a^{31}-\frac{1}{14}a^{30}-\frac{1}{7}a^{29}+\frac{5}{14}a^{28}+\frac{1}{14}a^{26}-\frac{1}{7}a^{25}+\frac{3}{7}a^{24}-\frac{2}{7}a^{22}-\frac{1}{7}a^{21}+\frac{2}{7}a^{20}-\frac{1}{7}a^{19}+\frac{3}{7}a^{18}-\frac{3}{7}a^{17}-\frac{3}{7}a^{16}-\frac{2}{7}a^{15}-\frac{1}{2}a^{14}+\frac{2}{7}a^{13}-\frac{3}{7}a^{12}-\frac{1}{14}a^{11}+\frac{2}{7}a^{9}+\frac{1}{14}a^{8}+\frac{3}{14}a^{7}-\frac{3}{14}a^{6}-\frac{1}{14}a^{5}+\frac{5}{14}a^{4}-\frac{1}{7}a^{3}+\frac{3}{14}a^{2}-\frac{1}{7}a+\frac{3}{7}$, $\frac{1}{14}a^{38}+\frac{3}{14}a^{35}+\frac{5}{14}a^{33}+\frac{3}{14}a^{32}-\frac{1}{14}a^{31}+\frac{3}{7}a^{30}+\frac{1}{14}a^{29}+\frac{5}{14}a^{27}+\frac{1}{7}a^{26}-\frac{3}{7}a^{25}+\frac{3}{7}a^{24}+\frac{2}{7}a^{23}+\frac{3}{7}a^{22}+\frac{3}{7}a^{21}+\frac{2}{7}a^{20}+\frac{2}{7}a^{19}-\frac{2}{7}a^{18}-\frac{1}{7}a^{17}-\frac{1}{7}a^{16}-\frac{1}{2}a^{15}-\frac{1}{7}a^{14}+\frac{1}{7}a^{13}-\frac{1}{14}a^{12}-\frac{1}{7}a^{11}+\frac{2}{7}a^{10}+\frac{1}{14}a^{9}-\frac{5}{14}a^{8}+\frac{1}{14}a^{7}-\frac{3}{14}a^{6}-\frac{5}{14}a^{5}+\frac{2}{7}a^{4}-\frac{1}{14}a^{3}-\frac{1}{7}a^{2}-\frac{1}{7}$, $\frac{1}{98}a^{39}+\frac{3}{98}a^{38}-\frac{1}{98}a^{37}-\frac{1}{49}a^{36}-\frac{5}{98}a^{35}-\frac{18}{49}a^{34}-\frac{1}{14}a^{33}-\frac{27}{98}a^{32}+\frac{1}{14}a^{31}+\frac{47}{98}a^{30}+\frac{8}{49}a^{29}-\frac{3}{7}a^{28}+\frac{24}{49}a^{27}-\frac{5}{98}a^{26}-\frac{33}{98}a^{25}-\frac{16}{49}a^{24}-\frac{23}{49}a^{23}-\frac{11}{49}a^{22}+\frac{18}{49}a^{21}+\frac{24}{49}a^{20}-\frac{1}{49}a^{19}-\frac{18}{49}a^{18}-\frac{17}{49}a^{17}-\frac{23}{98}a^{16}+\frac{9}{98}a^{15}+\frac{23}{98}a^{14}-\frac{1}{7}a^{13}+\frac{29}{98}a^{12}+\frac{43}{98}a^{11}-\frac{18}{49}a^{10}-\frac{17}{49}a^{9}-\frac{3}{14}a^{8}+\frac{1}{7}a^{7}-\frac{8}{49}a^{6}-\frac{5}{14}a^{5}-\frac{3}{14}a^{4}-\frac{24}{49}a^{3}-\frac{9}{98}a^{2}+\frac{13}{98}a-\frac{12}{49}$, $\frac{1}{7641844}a^{40}-\frac{8037}{7641844}a^{39}+\frac{53629}{1910461}a^{38}+\frac{83161}{3820922}a^{37}-\frac{43773}{3820922}a^{36}-\frac{362625}{1910461}a^{35}-\frac{200847}{7641844}a^{34}+\frac{1393743}{7641844}a^{33}-\frac{3548199}{7641844}a^{32}-\frac{386115}{7641844}a^{31}-\frac{1020917}{3820922}a^{30}+\frac{1302615}{7641844}a^{29}-\frac{3812453}{7641844}a^{28}-\frac{1495719}{7641844}a^{27}+\frac{625102}{1910461}a^{26}+\frac{1611871}{7641844}a^{25}-\frac{660991}{7641844}a^{24}-\frac{859981}{3820922}a^{23}-\frac{1505743}{3820922}a^{22}-\frac{1304281}{3820922}a^{21}+\frac{39456}{1910461}a^{20}+\frac{109703}{545846}a^{19}-\frac{1248787}{3820922}a^{18}+\frac{650413}{7641844}a^{17}+\frac{217451}{7641844}a^{16}-\frac{15641}{3820922}a^{15}-\frac{1290307}{3820922}a^{14}-\frac{713317}{3820922}a^{13}+\frac{2938709}{7641844}a^{12}+\frac{614674}{1910461}a^{11}-\frac{2755573}{7641844}a^{10}-\frac{2411945}{7641844}a^{9}+\frac{378243}{1091692}a^{8}-\frac{460723}{1910461}a^{7}+\frac{13274}{1910461}a^{6}+\frac{24442}{272923}a^{5}+\frac{395133}{3820922}a^{4}+\frac{650519}{1910461}a^{3}+\frac{37823}{545846}a^{2}+\frac{352689}{7641844}a+\frac{382233}{7641844}$, $\frac{1}{1918102844}a^{41}+\frac{13}{1918102844}a^{40}-\frac{4481499}{959051422}a^{39}-\frac{1985485}{137007346}a^{38}-\frac{9006323}{959051422}a^{37}+\frac{11300052}{479525711}a^{36}+\frac{329936703}{1918102844}a^{35}+\frac{398903089}{1918102844}a^{34}+\frac{228202971}{1918102844}a^{33}+\frac{254395549}{1918102844}a^{32}-\frac{85161245}{479525711}a^{31}-\frac{943493847}{1918102844}a^{30}+\frac{126954705}{274014692}a^{29}-\frac{866171083}{1918102844}a^{28}-\frac{475328817}{959051422}a^{27}-\frac{49796415}{1918102844}a^{26}+\frac{889558207}{1918102844}a^{25}-\frac{196269231}{959051422}a^{24}-\frac{380439033}{959051422}a^{23}-\frac{394956143}{959051422}a^{22}+\frac{211731595}{479525711}a^{21}+\frac{205415005}{959051422}a^{20}+\frac{247325199}{959051422}a^{19}-\frac{399623731}{1918102844}a^{18}-\frac{69849529}{274014692}a^{17}+\frac{16832077}{68503673}a^{16}-\frac{8056619}{137007346}a^{15}+\frac{225043155}{959051422}a^{14}+\frac{797886625}{1918102844}a^{13}-\frac{5108784}{479525711}a^{12}+\frac{290880459}{1918102844}a^{11}+\frac{922134233}{1918102844}a^{10}+\frac{34281629}{1918102844}a^{9}+\frac{158910653}{959051422}a^{8}-\frac{220850936}{479525711}a^{7}-\frac{434528427}{959051422}a^{6}-\frac{162463761}{479525711}a^{5}+\frac{137665623}{959051422}a^{4}-\frac{237654654}{479525711}a^{3}-\frac{4684525}{39144956}a^{2}-\frac{457225201}{1918102844}a+\frac{477851}{1910461}$, $\frac{1}{1452003852908}a^{42}-\frac{289}{1452003852908}a^{41}-\frac{7078}{363000963227}a^{40}+\frac{2000663363}{726001926454}a^{39}+\frac{1766007249}{363000963227}a^{38}-\frac{1534872873}{363000963227}a^{37}-\frac{47252113721}{1452003852908}a^{36}-\frac{1459268155}{1452003852908}a^{35}+\frac{437156211241}{1452003852908}a^{34}-\frac{21345434387}{1452003852908}a^{33}-\frac{151300259791}{363000963227}a^{32}-\frac{309618896565}{1452003852908}a^{31}-\frac{32193270701}{207429121844}a^{30}-\frac{292264881011}{1452003852908}a^{29}-\frac{166228892527}{363000963227}a^{28}-\frac{584655883509}{1452003852908}a^{27}-\frac{96424053853}{207429121844}a^{26}-\frac{176610905602}{363000963227}a^{25}+\frac{32705817849}{726001926454}a^{24}+\frac{240853080485}{726001926454}a^{23}-\frac{14471632795}{51857280461}a^{22}-\frac{689743445}{2892437954}a^{21}+\frac{139114695861}{726001926454}a^{20}-\frac{47876046901}{207429121844}a^{19}+\frac{539119958155}{1452003852908}a^{18}-\frac{73069425077}{726001926454}a^{17}+\frac{246285556621}{726001926454}a^{16}+\frac{22058670919}{363000963227}a^{15}-\frac{84163912107}{1452003852908}a^{14}-\frac{197368246905}{726001926454}a^{13}-\frac{500492783303}{1452003852908}a^{12}-\frac{8026334997}{1452003852908}a^{11}-\frac{227848774205}{1452003852908}a^{10}+\frac{328133469583}{726001926454}a^{9}+\frac{109666746019}{726001926454}a^{8}+\frac{120163519519}{363000963227}a^{7}-\frac{124130969043}{726001926454}a^{6}+\frac{9832296632}{51857280461}a^{5}-\frac{3216877586}{51857280461}a^{4}-\frac{85316921609}{207429121844}a^{3}+\frac{87907318045}{1452003852908}a^{2}-\frac{72516168453}{726001926454}a-\frac{920067}{1910461}$, $\frac{1}{651949729955692}a^{43}-\frac{1}{651949729955692}a^{42}-\frac{9595}{325974864977846}a^{41}-\frac{5340723}{93135675707956}a^{40}+\frac{2613248973595}{651949729955692}a^{39}-\frac{1255618181691}{46567837853978}a^{38}-\frac{22262358366341}{651949729955692}a^{37}+\frac{13597849861119}{651949729955692}a^{36}+\frac{116602283151177}{651949729955692}a^{35}-\frac{94276757531137}{325974864977846}a^{34}+\frac{320818976260565}{651949729955692}a^{33}+\frac{160956542061969}{325974864977846}a^{32}+\frac{68213595294505}{325974864977846}a^{31}-\frac{999176780939}{93135675707956}a^{30}-\frac{209590715691389}{651949729955692}a^{29}-\frac{110638720947427}{325974864977846}a^{28}+\frac{53954304058138}{162987432488923}a^{27}-\frac{5901219716860}{23283918926989}a^{26}-\frac{2578107665745}{13305096529708}a^{25}+\frac{314209182010113}{651949729955692}a^{24}-\frac{40395889895561}{325974864977846}a^{23}+\frac{11844545302956}{162987432488923}a^{22}-\frac{5804834786019}{162987432488923}a^{21}+\frac{24838206695}{53008352708}a^{20}+\frac{4169336828137}{651949729955692}a^{19}+\frac{38629661480511}{325974864977846}a^{18}+\frac{9980430330435}{651949729955692}a^{17}+\frac{47405346957851}{651949729955692}a^{16}-\frac{320625290663}{1452003852908}a^{15}+\frac{36049741487765}{325974864977846}a^{14}-\frac{289577273614331}{651949729955692}a^{13}+\frac{22487900234517}{162987432488923}a^{12}-\frac{240230313041545}{651949729955692}a^{11}-\frac{286539712536439}{651949729955692}a^{10}-\frac{16645323673957}{93135675707956}a^{9}+\frac{29664969831141}{651949729955692}a^{8}+\frac{73094012878195}{325974864977846}a^{7}-\frac{49154764562366}{162987432488923}a^{6}-\frac{120112930571489}{325974864977846}a^{5}+\frac{249724968142697}{651949729955692}a^{4}-\frac{309435012219303}{651949729955692}a^{3}-\frac{67784243258591}{162987432488923}a^{2}+\frac{200741499826307}{651949729955692}a+\frac{1163162417}{3431187956}$, $\frac{1}{96\!\cdots\!08}a^{44}-\frac{22\!\cdots\!10}{24\!\cdots\!77}a^{43}+\frac{29\!\cdots\!59}{96\!\cdots\!08}a^{42}+\frac{49\!\cdots\!99}{48\!\cdots\!54}a^{41}+\frac{83\!\cdots\!51}{24\!\cdots\!77}a^{40}-\frac{11\!\cdots\!32}{24\!\cdots\!77}a^{39}-\frac{13\!\cdots\!71}{96\!\cdots\!08}a^{38}-\frac{19\!\cdots\!01}{24\!\cdots\!77}a^{37}-\frac{85\!\cdots\!74}{24\!\cdots\!77}a^{36}-\frac{25\!\cdots\!65}{48\!\cdots\!54}a^{35}+\frac{39\!\cdots\!33}{96\!\cdots\!08}a^{34}+\frac{13\!\cdots\!89}{96\!\cdots\!08}a^{33}+\frac{70\!\cdots\!46}{24\!\cdots\!77}a^{32}-\frac{23\!\cdots\!73}{48\!\cdots\!54}a^{31}-\frac{33\!\cdots\!71}{96\!\cdots\!08}a^{30}-\frac{17\!\cdots\!15}{96\!\cdots\!08}a^{29}-\frac{25\!\cdots\!15}{68\!\cdots\!22}a^{28}+\frac{37\!\cdots\!09}{96\!\cdots\!08}a^{27}-\frac{17\!\cdots\!25}{24\!\cdots\!77}a^{26}+\frac{27\!\cdots\!31}{68\!\cdots\!22}a^{25}-\frac{52\!\cdots\!51}{48\!\cdots\!54}a^{24}-\frac{47\!\cdots\!55}{48\!\cdots\!54}a^{23}-\frac{56\!\cdots\!79}{24\!\cdots\!77}a^{22}+\frac{42\!\cdots\!49}{13\!\cdots\!44}a^{21}+\frac{42\!\cdots\!45}{24\!\cdots\!77}a^{20}+\frac{14\!\cdots\!49}{96\!\cdots\!08}a^{19}+\frac{66\!\cdots\!41}{24\!\cdots\!77}a^{18}-\frac{42\!\cdots\!31}{24\!\cdots\!77}a^{17}+\frac{46\!\cdots\!73}{96\!\cdots\!08}a^{16}+\frac{61\!\cdots\!69}{96\!\cdots\!08}a^{15}+\frac{47\!\cdots\!51}{96\!\cdots\!08}a^{14}-\frac{10\!\cdots\!07}{68\!\cdots\!22}a^{13}-\frac{11\!\cdots\!99}{24\!\cdots\!77}a^{12}-\frac{36\!\cdots\!67}{96\!\cdots\!08}a^{11}-\frac{28\!\cdots\!76}{24\!\cdots\!77}a^{10}+\frac{64\!\cdots\!07}{48\!\cdots\!54}a^{9}+\frac{59\!\cdots\!30}{34\!\cdots\!11}a^{8}-\frac{26\!\cdots\!57}{37\!\cdots\!02}a^{7}+\frac{15\!\cdots\!33}{48\!\cdots\!54}a^{6}+\frac{17\!\cdots\!01}{96\!\cdots\!08}a^{5}-\frac{11\!\cdots\!67}{48\!\cdots\!54}a^{4}+\frac{24\!\cdots\!03}{96\!\cdots\!08}a^{3}-\frac{28\!\cdots\!30}{24\!\cdots\!77}a^{2}-\frac{38\!\cdots\!17}{68\!\cdots\!22}a+\frac{17\!\cdots\!13}{18\!\cdots\!73}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

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 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757)
 
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 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757, 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 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757);
 
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 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757);
 
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

$C_3\times C_{15}$ (as 45T2):

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)
 
An abelian group of order 45
The 45 conjugacy class representatives for $C_3\times C_{15}$
Character table for $C_3\times C_{15}$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, 5.5.390625.1, 9.9.62523502209.1, 15.15.16836836874485015869140625.1, 15.15.207828545629978179931640625.1, 15.15.58706420176135948240756988525390625.1, 15.15.58706420176135948240756988525390625.2

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 $15^{3}$ R R R $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$

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
\(3\) Copy content Toggle raw display Deg $45$$3$$15$$60$
\(5\) Copy content Toggle raw display 5.15.24.88$x^{15} + 60 x^{14} + 1200 x^{13} + 8000 x^{12} + 15 x^{10} + 600 x^{9} + 6000 x^{8} + 7575 x^{5} + 151500 x^{4} - 337375$$5$$3$$24$$C_{15}$$[2]^{3}$
5.15.24.88$x^{15} + 60 x^{14} + 1200 x^{13} + 8000 x^{12} + 15 x^{10} + 600 x^{9} + 6000 x^{8} + 7575 x^{5} + 151500 x^{4} - 337375$$5$$3$$24$$C_{15}$$[2]^{3}$
5.15.24.88$x^{15} + 60 x^{14} + 1200 x^{13} + 8000 x^{12} + 15 x^{10} + 600 x^{9} + 6000 x^{8} + 7575 x^{5} + 151500 x^{4} - 337375$$5$$3$$24$$C_{15}$$[2]^{3}$
\(7\) Copy content Toggle raw display 7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$