Properties

Label 45.45.162...641.1
Degree $45$
Signature $[45, 0]$
Discriminant $1.623\times 10^{109}$
Root discriminant \(267.24\)
Ramified primes $19,41$
Class number not computed
Class group not computed
Galois group $C_{45}$ (as 45T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187)
 
gp: K = bnfinit(y^45 - 4*y^44 - 182*y^43 + 580*y^42 + 15229*y^41 - 35784*y^40 - 771497*y^39 + 1192202*y^38 + 26249095*y^37 - 21282586*y^36 - 630350986*y^35 + 101314575*y^34 + 10957230656*y^33 + 4486413137*y^32 - 139417559409*y^31 - 128053969474*y^30 + 1299055132435*y^29 + 1816090021004*y^28 - 8766783201604*y^27 - 16658946907076*y^26 + 41634175647095*y^25 + 105419181053468*y^24 - 129801264303153*y^23 - 468263377533416*y^22 + 210417608171782*y^21 + 1454214352745320*y^20 + 113648989333176*y^19 - 3097196961798623*y^18 - 1450605309591570*y^17 + 4363534978783893*y^16 + 3500531720706214*y^15 - 3807203164617687*y^14 - 4497701352355764*y^13 + 1760587964112158*y^12 + 3382378542501223*y^11 - 165688879950870*y^10 - 1480388191573288*y^9 - 205996027224400*y^8 + 358811476146228*y^7 + 88160595048203*y^6 - 42989403930482*y^5 - 13014696239433*y^4 + 1879419740489*y^3 + 646702182070*y^2 + 19352782133*y - 2280393187, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187)
 

\( x^{45} - 4 x^{44} - 182 x^{43} + 580 x^{42} + 15229 x^{41} - 35784 x^{40} - 771497 x^{39} + 1192202 x^{38} + 26249095 x^{37} - 21282586 x^{36} - 630350986 x^{35} + \cdots - 2280393187 \) 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:   \(162\!\cdots\!641\) \(\medspace = 19^{40}\cdot 41^{36}\) 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:  \(267.24\)
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))
 
Ramified primes:   \(19\), \(41\) 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:  \(779=19\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{779}(256,·)$, $\chi_{779}(1,·)$, $\chi_{779}(387,·)$, $\chi_{779}(385,·)$, $\chi_{779}(264,·)$, $\chi_{779}(139,·)$, $\chi_{779}(652,·)$, $\chi_{779}(16,·)$, $\chi_{779}(529,·)$, $\chi_{779}(406,·)$, $\chi_{779}(666,·)$, $\chi_{779}(283,·)$, $\chi_{779}(543,·)$, $\chi_{779}(672,·)$, $\chi_{779}(674,·)$, $\chi_{779}(549,·)$, $\chi_{779}(42,·)$, $\chi_{779}(305,·)$, $\chi_{779}(180,·)$, $\chi_{779}(693,·)$, $\chi_{779}(182,·)$, $\chi_{779}(329,·)$, $\chi_{779}(575,·)$, $\chi_{779}(707,·)$, $\chi_{779}(324,·)$, $\chi_{779}(201,·)$, $\chi_{779}(631,·)$, $\chi_{779}(461,·)$, $\chi_{779}(590,·)$, $\chi_{779}(83,·)$, $\chi_{779}(206,·)$, $\chi_{779}(215,·)$, $\chi_{779}(346,·)$, $\chi_{779}(92,·)$, $\chi_{779}(739,·)$, $\chi_{779}(100,·)$, $\chi_{779}(657,·)$, $\chi_{779}(748,·)$, $\chi_{779}(365,·)$, $\chi_{779}(625,·)$, $\chi_{779}(370,·)$, $\chi_{779}(467,·)$, $\chi_{779}(119,·)$, $\chi_{779}(633,·)$, $\chi_{779}(510,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{3}a^{27}-\frac{1}{3}a^{21}+\frac{1}{3}a^{19}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{28}-\frac{1}{3}a^{22}+\frac{1}{3}a^{20}-\frac{1}{3}a^{16}+\frac{1}{3}a^{14}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{29}-\frac{1}{3}a^{23}+\frac{1}{3}a^{21}-\frac{1}{3}a^{17}+\frac{1}{3}a^{15}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{30}-\frac{1}{3}a^{24}+\frac{1}{3}a^{22}-\frac{1}{3}a^{18}+\frac{1}{3}a^{16}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{31}-\frac{1}{3}a^{25}+\frac{1}{3}a^{23}-\frac{1}{3}a^{19}+\frac{1}{3}a^{17}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{32}-\frac{1}{3}a^{26}+\frac{1}{3}a^{24}-\frac{1}{3}a^{20}+\frac{1}{3}a^{18}+\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{33}+\frac{1}{3}a^{25}+\frac{1}{3}a^{21}-\frac{1}{3}a^{19}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{34}+\frac{1}{3}a^{26}+\frac{1}{3}a^{22}-\frac{1}{3}a^{20}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{35}+\frac{1}{3}a^{23}-\frac{1}{3}a^{19}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{36}+\frac{1}{9}a^{35}-\frac{1}{9}a^{34}-\frac{1}{9}a^{33}+\frac{1}{9}a^{32}-\frac{1}{9}a^{31}-\frac{1}{9}a^{30}-\frac{1}{9}a^{29}-\frac{1}{9}a^{28}+\frac{1}{9}a^{26}-\frac{2}{9}a^{23}-\frac{1}{9}a^{22}+\frac{4}{9}a^{21}+\frac{4}{9}a^{20}-\frac{2}{9}a^{19}-\frac{1}{9}a^{18}+\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{4}{9}a^{14}-\frac{4}{9}a^{13}-\frac{1}{3}a^{12}-\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{2}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{3}a^{7}+\frac{2}{9}a^{6}+\frac{4}{9}a^{5}+\frac{2}{9}a^{4}-\frac{4}{9}a^{3}+\frac{4}{9}a^{2}+\frac{1}{9}a-\frac{1}{9}$, $\frac{1}{9}a^{37}+\frac{1}{9}a^{35}-\frac{1}{9}a^{33}+\frac{1}{9}a^{32}+\frac{1}{9}a^{28}+\frac{1}{9}a^{27}-\frac{4}{9}a^{26}-\frac{1}{3}a^{25}+\frac{1}{9}a^{24}+\frac{4}{9}a^{23}-\frac{4}{9}a^{22}-\frac{1}{3}a^{21}+\frac{1}{9}a^{19}-\frac{2}{9}a^{18}+\frac{1}{3}a^{16}-\frac{2}{9}a^{15}-\frac{2}{9}a^{14}+\frac{1}{9}a^{13}-\frac{4}{9}a^{12}-\frac{4}{9}a^{11}+\frac{1}{9}a^{10}-\frac{4}{9}a^{9}-\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{4}{9}a^{6}+\frac{1}{9}a^{5}-\frac{1}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{9}a^{38}-\frac{1}{9}a^{35}-\frac{1}{9}a^{33}-\frac{1}{9}a^{32}+\frac{1}{9}a^{31}+\frac{1}{9}a^{30}-\frac{1}{9}a^{29}-\frac{1}{9}a^{28}-\frac{1}{9}a^{27}-\frac{4}{9}a^{26}-\frac{2}{9}a^{25}+\frac{4}{9}a^{24}+\frac{1}{9}a^{23}+\frac{1}{9}a^{22}-\frac{4}{9}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{19}+\frac{1}{9}a^{18}+\frac{1}{3}a^{17}-\frac{2}{9}a^{16}+\frac{4}{9}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{4}{9}a^{12}-\frac{1}{9}a^{11}-\frac{2}{9}a^{10}-\frac{1}{3}a^{9}-\frac{2}{9}a^{8}+\frac{2}{9}a^{7}-\frac{1}{9}a^{6}-\frac{4}{9}a^{5}-\frac{1}{3}a^{4}+\frac{4}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{9}$, $\frac{1}{9}a^{39}+\frac{1}{9}a^{35}+\frac{1}{9}a^{34}+\frac{1}{9}a^{33}-\frac{1}{9}a^{32}+\frac{1}{9}a^{30}+\frac{1}{9}a^{29}+\frac{1}{9}a^{28}-\frac{1}{9}a^{27}-\frac{4}{9}a^{26}-\frac{2}{9}a^{25}+\frac{4}{9}a^{24}-\frac{4}{9}a^{23}-\frac{2}{9}a^{22}+\frac{1}{9}a^{21}+\frac{4}{9}a^{20}-\frac{1}{9}a^{19}-\frac{4}{9}a^{18}-\frac{2}{9}a^{17}-\frac{2}{9}a^{16}-\frac{2}{9}a^{14}+\frac{4}{9}a^{13}-\frac{1}{9}a^{12}-\frac{2}{9}a^{10}+\frac{1}{3}a^{8}-\frac{1}{9}a^{7}-\frac{2}{9}a^{6}+\frac{1}{9}a^{5}+\frac{1}{3}a^{4}+\frac{2}{9}a^{3}+\frac{4}{9}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{5139}a^{40}+\frac{3}{571}a^{39}+\frac{157}{5139}a^{38}+\frac{11}{571}a^{37}-\frac{25}{571}a^{36}-\frac{712}{5139}a^{35}-\frac{394}{5139}a^{34}+\frac{782}{5139}a^{33}+\frac{223}{5139}a^{32}-\frac{61}{571}a^{31}+\frac{13}{571}a^{30}+\frac{319}{5139}a^{29}-\frac{499}{5139}a^{28}-\frac{143}{5139}a^{27}-\frac{2380}{5139}a^{26}-\frac{742}{5139}a^{25}+\frac{637}{1713}a^{24}-\frac{239}{5139}a^{23}-\frac{279}{571}a^{22}+\frac{146}{5139}a^{21}-\frac{77}{5139}a^{20}+\frac{2506}{5139}a^{19}+\frac{643}{1713}a^{18}-\frac{14}{5139}a^{17}-\frac{1358}{5139}a^{16}+\frac{1616}{5139}a^{15}-\frac{125}{571}a^{14}-\frac{718}{1713}a^{13}+\frac{1835}{5139}a^{12}-\frac{2549}{5139}a^{11}-\frac{188}{1713}a^{10}+\frac{1948}{5139}a^{9}+\frac{449}{5139}a^{8}-\frac{856}{1713}a^{7}-\frac{1859}{5139}a^{6}-\frac{1634}{5139}a^{5}-\frac{64}{1713}a^{4}+\frac{115}{1713}a^{3}-\frac{1049}{5139}a^{2}-\frac{449}{5139}a-\frac{616}{5139}$, $\frac{1}{5139}a^{41}-\frac{1}{5139}a^{39}-\frac{143}{5139}a^{38}-\frac{43}{5139}a^{37}+\frac{224}{5139}a^{36}-\frac{584}{5139}a^{35}+\frac{236}{5139}a^{33}+\frac{94}{1713}a^{32}+\frac{94}{5139}a^{31}+\frac{5}{1713}a^{30}-\frac{547}{5139}a^{29}+\frac{256}{1713}a^{28}-\frac{232}{5139}a^{27}-\frac{2147}{5139}a^{26}-\frac{895}{5139}a^{25}+\frac{125}{5139}a^{24}-\frac{1768}{5139}a^{23}+\frac{1136}{5139}a^{22}+\frac{1120}{5139}a^{21}+\frac{17}{5139}a^{20}+\frac{1645}{5139}a^{19}+\frac{145}{1713}a^{18}-\frac{409}{5139}a^{17}+\frac{1738}{5139}a^{16}-\frac{644}{1713}a^{15}+\frac{842}{1713}a^{14}-\frac{1675}{5139}a^{13}+\frac{239}{571}a^{12}+\frac{484}{1713}a^{11}+\frac{2330}{5139}a^{10}-\frac{62}{1713}a^{9}-\frac{1558}{5139}a^{8}+\frac{11}{571}a^{7}-\frac{547}{5139}a^{6}+\frac{530}{5139}a^{5}-\frac{147}{571}a^{4}+\frac{485}{5139}a^{3}-\frac{2389}{5139}a^{2}-\frac{542}{1713}a+\frac{644}{5139}$, $\frac{1}{981549}a^{42}+\frac{3}{109061}a^{41}+\frac{74}{981549}a^{40}+\frac{46964}{981549}a^{39}-\frac{46945}{981549}a^{38}-\frac{14777}{327183}a^{37}+\frac{7426}{327183}a^{36}-\frac{142256}{981549}a^{35}-\frac{89840}{981549}a^{34}+\frac{59023}{981549}a^{33}-\frac{12385}{109061}a^{32}-\frac{31199}{981549}a^{31}-\frac{69023}{981549}a^{30}+\frac{137828}{981549}a^{29}+\frac{1922}{981549}a^{28}+\frac{49384}{981549}a^{27}+\frac{161765}{981549}a^{26}+\frac{14168}{327183}a^{25}+\frac{207742}{981549}a^{24}-\frac{265517}{981549}a^{23}+\frac{190064}{981549}a^{22}-\frac{462415}{981549}a^{21}+\frac{109387}{981549}a^{20}+\frac{156857}{981549}a^{19}-\frac{15720}{109061}a^{18}-\frac{381505}{981549}a^{17}+\frac{130432}{981549}a^{16}+\frac{488392}{981549}a^{15}+\frac{33863}{109061}a^{14}-\frac{153287}{327183}a^{13}+\frac{288514}{981549}a^{12}+\frac{232358}{981549}a^{11}+\frac{158035}{981549}a^{10}+\frac{386192}{981549}a^{9}-\frac{62537}{981549}a^{8}+\frac{99434}{327183}a^{7}-\frac{24243}{109061}a^{6}-\frac{213485}{981549}a^{5}+\frac{41180}{109061}a^{4}-\frac{354554}{981549}a^{3}+\frac{4264}{327183}a^{2}+\frac{42406}{981549}a+\frac{91669}{981549}$, $\frac{1}{635062203}a^{43}-\frac{113}{635062203}a^{42}+\frac{14057}{635062203}a^{41}-\frac{37504}{635062203}a^{40}-\frac{511703}{70562467}a^{39}+\frac{2358713}{211687401}a^{38}-\frac{32408963}{635062203}a^{37}-\frac{8891665}{635062203}a^{36}+\frac{20228819}{635062203}a^{35}+\frac{98001590}{635062203}a^{34}-\frac{33253706}{211687401}a^{33}-\frac{6412873}{635062203}a^{32}+\frac{510347}{3324933}a^{31}+\frac{87336925}{635062203}a^{30}+\frac{6406796}{211687401}a^{29}+\frac{74722783}{635062203}a^{28}+\frac{1580413}{211687401}a^{27}+\frac{253270265}{635062203}a^{26}+\frac{68465075}{635062203}a^{25}+\frac{40320527}{635062203}a^{24}-\frac{143848424}{635062203}a^{23}-\frac{164349569}{635062203}a^{22}-\frac{121405972}{635062203}a^{21}+\frac{85734034}{635062203}a^{20}-\frac{12536910}{70562467}a^{19}+\frac{59954213}{211687401}a^{18}+\frac{5269182}{70562467}a^{17}-\frac{240818641}{635062203}a^{16}-\frac{164671610}{635062203}a^{15}+\frac{73225948}{635062203}a^{14}+\frac{4978212}{70562467}a^{13}-\frac{223988698}{635062203}a^{12}-\frac{152169004}{635062203}a^{11}+\frac{267789406}{635062203}a^{10}-\frac{134177899}{635062203}a^{9}-\frac{16454174}{70562467}a^{8}-\frac{34711259}{211687401}a^{7}+\frac{122065793}{635062203}a^{6}-\frac{1416741}{70562467}a^{5}+\frac{41805829}{211687401}a^{4}+\frac{7389673}{70562467}a^{3}+\frac{199109609}{635062203}a^{2}-\frac{80971457}{211687401}a-\frac{167540222}{635062203}$, $\frac{1}{69\!\cdots\!33}a^{44}+\frac{57\!\cdots\!46}{69\!\cdots\!33}a^{43}-\frac{67\!\cdots\!70}{23\!\cdots\!11}a^{42}+\frac{21\!\cdots\!75}{23\!\cdots\!11}a^{41}-\frac{17\!\cdots\!10}{23\!\cdots\!11}a^{40}+\frac{79\!\cdots\!88}{23\!\cdots\!11}a^{39}-\frac{17\!\cdots\!78}{23\!\cdots\!11}a^{38}+\frac{90\!\cdots\!26}{77\!\cdots\!37}a^{37}+\frac{24\!\cdots\!74}{77\!\cdots\!37}a^{36}+\frac{22\!\cdots\!72}{69\!\cdots\!33}a^{35}-\frac{40\!\cdots\!67}{69\!\cdots\!33}a^{34}-\frac{27\!\cdots\!76}{69\!\cdots\!33}a^{33}-\frac{44\!\cdots\!79}{23\!\cdots\!11}a^{32}+\frac{70\!\cdots\!88}{69\!\cdots\!33}a^{31}-\frac{39\!\cdots\!99}{69\!\cdots\!33}a^{30}+\frac{79\!\cdots\!49}{69\!\cdots\!33}a^{29}+\frac{10\!\cdots\!40}{69\!\cdots\!33}a^{28}+\frac{96\!\cdots\!99}{69\!\cdots\!33}a^{27}+\frac{19\!\cdots\!03}{77\!\cdots\!37}a^{26}-\frac{26\!\cdots\!44}{69\!\cdots\!33}a^{25}-\frac{97\!\cdots\!02}{69\!\cdots\!33}a^{24}-\frac{17\!\cdots\!56}{69\!\cdots\!33}a^{23}+\frac{29\!\cdots\!90}{69\!\cdots\!33}a^{22}+\frac{28\!\cdots\!10}{69\!\cdots\!33}a^{21}-\frac{16\!\cdots\!37}{23\!\cdots\!11}a^{20}-\frac{14\!\cdots\!95}{69\!\cdots\!33}a^{19}+\frac{19\!\cdots\!19}{69\!\cdots\!33}a^{18}-\frac{99\!\cdots\!55}{23\!\cdots\!11}a^{17}+\frac{16\!\cdots\!10}{69\!\cdots\!33}a^{16}+\frac{29\!\cdots\!53}{69\!\cdots\!33}a^{15}-\frac{50\!\cdots\!42}{23\!\cdots\!11}a^{14}+\frac{77\!\cdots\!55}{77\!\cdots\!37}a^{13}-\frac{29\!\cdots\!72}{23\!\cdots\!11}a^{12}-\frac{50\!\cdots\!75}{69\!\cdots\!33}a^{11}+\frac{19\!\cdots\!18}{69\!\cdots\!33}a^{10}-\frac{42\!\cdots\!51}{23\!\cdots\!11}a^{9}-\frac{45\!\cdots\!53}{69\!\cdots\!33}a^{8}-\frac{13\!\cdots\!77}{69\!\cdots\!33}a^{7}-\frac{18\!\cdots\!80}{69\!\cdots\!33}a^{6}-\frac{24\!\cdots\!32}{69\!\cdots\!33}a^{5}+\frac{12\!\cdots\!21}{69\!\cdots\!33}a^{4}+\frac{44\!\cdots\!70}{23\!\cdots\!11}a^{3}-\frac{24\!\cdots\!40}{69\!\cdots\!33}a^{2}-\frac{10\!\cdots\!10}{23\!\cdots\!11}a+\frac{14\!\cdots\!00}{69\!\cdots\!33}$ 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 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187)
 
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 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187, 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 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187);
 
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 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187);
 
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

3.3.361.1, 5.5.2825761.1, \(\Q(\zeta_{19})^+\), 15.15.138338254038795273955595483867881.1

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 $45$ ${\href{/padicField/3.9.0.1}{9} }^{5}$ $45$ $15^{3}$ $15^{3}$ $45$ $45$ R $45$ $45$ $15^{3}$ ${\href{/padicField/37.5.0.1}{5} }^{9}$ R $45$ $45$ $45$ $45$

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
\(19\) Copy content Toggle raw display Deg $45$$9$$5$$40$
\(41\) Copy content Toggle raw display Deg $45$$5$$9$$36$