Properties

Label 45.45.408...321.1
Degree $45$
Signature $[45, 0]$
Discriminant $4.084\times 10^{103}$
Root discriminant \(200.66\)
Ramified primes $3,61$
Class number not computed
Class group not computed
Galois group $C_3\times C_{15}$ (as 45T2)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289)
 
gp: K = bnfinit(y^45 - 3*y^44 - 126*y^43 + 350*y^42 + 7095*y^41 - 18207*y^40 - 237254*y^39 + 560949*y^38 + 5280957*y^37 - 11465163*y^36 - 83127609*y^35 + 164960568*y^34 + 959694733*y^33 - 1729668720*y^32 - 8316855543*y^31 + 13495839126*y^30 + 54911329452*y^29 - 79318322604*y^28 - 278624618803*y^27 + 353264598507*y^26 + 1090386630918*y^25 - 1193881051406*y^24 - 3286918329246*y^23 + 3054422504946*y^22 + 7584376524784*y^21 - 5885779838802*y^20 - 13239515051214*y^19 + 8489225140885*y^18 + 17170281215082*y^17 - 9118144480932*y^16 - 16115327765898*y^15 + 7281918623109*y^14 + 10535410010724*y^13 - 4323241543716*y^12 - 4518883017183*y^11 + 1877328461526*y^10 + 1138556965481*y^9 - 554295751305*y^8 - 124509763362*y^7 + 91825606123*y^6 - 3656082528*y^5 - 5240896776*y^4 + 961503906*y^3 - 1541688*y^2 - 11847522*y + 756289, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289)
 

\( x^{45} - 3 x^{44} - 126 x^{43} + 350 x^{42} + 7095 x^{41} - 18207 x^{40} - 237254 x^{39} + 560949 x^{38} + 5280957 x^{37} - 11465163 x^{36} - 83127609 x^{35} + \cdots + 756289 \) 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:   \(408\!\cdots\!321\) \(\medspace = 3^{60}\cdot 61^{42}\) 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:  \(200.66\)
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:   \(3\), \(61\) 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:  \(549=3^{2}\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{549}(256,·)$, $\chi_{549}(1,·)$, $\chi_{549}(259,·)$, $\chi_{549}(388,·)$, $\chi_{549}(391,·)$, $\chi_{549}(13,·)$, $\chi_{549}(142,·)$, $\chi_{549}(16,·)$, $\chi_{549}(22,·)$, $\chi_{549}(535,·)$, $\chi_{549}(25,·)$, $\chi_{549}(286,·)$, $\chi_{549}(544,·)$, $\chi_{549}(34,·)$, $\chi_{549}(424,·)$, $\chi_{549}(169,·)$, $\chi_{549}(301,·)$, $\chi_{549}(178,·)$, $\chi_{549}(436,·)$, $\chi_{549}(439,·)$, $\chi_{549}(184,·)$, $\chi_{549}(58,·)$, $\chi_{549}(196,·)$, $\chi_{549}(325,·)$, $\chi_{549}(70,·)$, $\chi_{549}(199,·)$, $\chi_{549}(73,·)$, $\chi_{549}(76,·)$, $\chi_{549}(205,·)$, $\chi_{549}(208,·)$, $\chi_{549}(469,·)$, $\chi_{549}(217,·)$, $\chi_{549}(442,·)$, $\chi_{549}(352,·)$, $\chi_{549}(400,·)$, $\chi_{549}(484,·)$, $\chi_{549}(103,·)$, $\chi_{549}(361,·)$, $\chi_{549}(367,·)$, $\chi_{549}(241,·)$, $\chi_{549}(118,·)$, $\chi_{549}(379,·)$, $\chi_{549}(508,·)$, $\chi_{549}(253,·)$, $\chi_{549}(382,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{11}a^{27}-\frac{1}{11}a^{26}+\frac{3}{11}a^{25}+\frac{3}{11}a^{24}-\frac{1}{11}a^{23}+\frac{5}{11}a^{22}+\frac{3}{11}a^{21}+\frac{4}{11}a^{20}+\frac{4}{11}a^{19}+\frac{3}{11}a^{18}-\frac{5}{11}a^{17}-\frac{5}{11}a^{16}+\frac{1}{11}a^{15}+\frac{3}{11}a^{13}-\frac{5}{11}a^{12}+\frac{2}{11}a^{11}-\frac{1}{11}a^{10}-\frac{5}{11}a^{9}-\frac{1}{11}a^{8}+\frac{4}{11}a^{7}+\frac{3}{11}a^{6}+\frac{4}{11}a^{5}+\frac{1}{11}a^{4}+\frac{1}{11}a^{3}+\frac{5}{11}a^{2}+\frac{5}{11}a+\frac{5}{11}$, $\frac{1}{11}a^{28}+\frac{2}{11}a^{26}-\frac{5}{11}a^{25}+\frac{2}{11}a^{24}+\frac{4}{11}a^{23}-\frac{3}{11}a^{22}-\frac{4}{11}a^{21}-\frac{3}{11}a^{20}-\frac{4}{11}a^{19}-\frac{2}{11}a^{18}+\frac{1}{11}a^{17}-\frac{4}{11}a^{16}+\frac{1}{11}a^{15}+\frac{3}{11}a^{14}-\frac{2}{11}a^{13}-\frac{3}{11}a^{12}+\frac{1}{11}a^{11}+\frac{5}{11}a^{10}+\frac{5}{11}a^{9}+\frac{3}{11}a^{8}-\frac{4}{11}a^{7}-\frac{4}{11}a^{6}+\frac{5}{11}a^{5}+\frac{2}{11}a^{4}-\frac{5}{11}a^{3}-\frac{1}{11}a^{2}-\frac{1}{11}a+\frac{5}{11}$, $\frac{1}{11}a^{29}-\frac{3}{11}a^{26}-\frac{4}{11}a^{25}-\frac{2}{11}a^{24}-\frac{1}{11}a^{23}-\frac{3}{11}a^{22}+\frac{2}{11}a^{21}-\frac{1}{11}a^{20}+\frac{1}{11}a^{19}-\frac{5}{11}a^{18}-\frac{5}{11}a^{17}+\frac{1}{11}a^{15}-\frac{2}{11}a^{14}+\frac{2}{11}a^{13}+\frac{1}{11}a^{11}-\frac{4}{11}a^{10}+\frac{2}{11}a^{9}-\frac{2}{11}a^{8}-\frac{1}{11}a^{7}-\frac{1}{11}a^{6}+\frac{5}{11}a^{5}+\frac{4}{11}a^{4}-\frac{3}{11}a^{3}-\frac{5}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{30}+\frac{4}{11}a^{26}-\frac{4}{11}a^{25}-\frac{3}{11}a^{24}+\frac{5}{11}a^{23}-\frac{5}{11}a^{22}-\frac{3}{11}a^{21}+\frac{2}{11}a^{20}-\frac{4}{11}a^{19}+\frac{4}{11}a^{18}-\frac{4}{11}a^{17}-\frac{3}{11}a^{16}+\frac{1}{11}a^{15}+\frac{2}{11}a^{14}-\frac{2}{11}a^{13}-\frac{3}{11}a^{12}+\frac{2}{11}a^{11}-\frac{1}{11}a^{10}+\frac{5}{11}a^{9}-\frac{4}{11}a^{8}+\frac{3}{11}a^{6}+\frac{5}{11}a^{5}+\frac{3}{11}a^{3}-\frac{1}{11}a^{2}+\frac{5}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{31}-\frac{4}{11}a^{25}+\frac{4}{11}a^{24}-\frac{1}{11}a^{23}-\frac{1}{11}a^{22}+\frac{1}{11}a^{21}+\frac{2}{11}a^{20}-\frac{1}{11}a^{19}-\frac{5}{11}a^{18}-\frac{5}{11}a^{17}-\frac{1}{11}a^{16}-\frac{2}{11}a^{15}-\frac{2}{11}a^{14}-\frac{4}{11}a^{13}+\frac{2}{11}a^{11}-\frac{2}{11}a^{10}+\frac{5}{11}a^{9}+\frac{4}{11}a^{8}-\frac{2}{11}a^{7}+\frac{4}{11}a^{6}-\frac{5}{11}a^{5}-\frac{1}{11}a^{4}-\frac{5}{11}a^{3}-\frac{4}{11}a^{2}-\frac{5}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{32}-\frac{4}{11}a^{26}+\frac{4}{11}a^{25}-\frac{1}{11}a^{24}-\frac{1}{11}a^{23}+\frac{1}{11}a^{22}+\frac{2}{11}a^{21}-\frac{1}{11}a^{20}-\frac{5}{11}a^{19}-\frac{5}{11}a^{18}-\frac{1}{11}a^{17}-\frac{2}{11}a^{16}-\frac{2}{11}a^{15}-\frac{4}{11}a^{14}+\frac{2}{11}a^{12}-\frac{2}{11}a^{11}+\frac{5}{11}a^{10}+\frac{4}{11}a^{9}-\frac{2}{11}a^{8}+\frac{4}{11}a^{7}-\frac{5}{11}a^{6}-\frac{1}{11}a^{5}-\frac{5}{11}a^{4}-\frac{4}{11}a^{3}-\frac{5}{11}a^{2}+\frac{2}{11}a$, $\frac{1}{11}a^{33}-\frac{3}{11}a^{23}+\frac{3}{11}a^{13}+\frac{2}{11}a^{11}-\frac{1}{11}a^{3}-\frac{2}{11}a-\frac{2}{11}$, $\frac{1}{11}a^{34}-\frac{3}{11}a^{24}+\frac{3}{11}a^{14}+\frac{2}{11}a^{12}-\frac{1}{11}a^{4}-\frac{2}{11}a^{2}-\frac{2}{11}a$, $\frac{1}{11}a^{35}-\frac{3}{11}a^{25}+\frac{3}{11}a^{15}+\frac{2}{11}a^{13}-\frac{1}{11}a^{5}-\frac{2}{11}a^{3}-\frac{2}{11}a^{2}$, $\frac{1}{11}a^{36}-\frac{3}{11}a^{26}+\frac{3}{11}a^{16}+\frac{2}{11}a^{14}-\frac{1}{11}a^{6}-\frac{2}{11}a^{4}-\frac{2}{11}a^{3}$, $\frac{1}{11}a^{37}-\frac{3}{11}a^{26}-\frac{2}{11}a^{25}-\frac{2}{11}a^{24}-\frac{3}{11}a^{23}+\frac{4}{11}a^{22}-\frac{2}{11}a^{21}+\frac{1}{11}a^{20}+\frac{1}{11}a^{19}-\frac{2}{11}a^{18}-\frac{1}{11}a^{17}-\frac{4}{11}a^{16}+\frac{5}{11}a^{15}-\frac{2}{11}a^{13}-\frac{4}{11}a^{12}-\frac{5}{11}a^{11}-\frac{3}{11}a^{10}-\frac{4}{11}a^{9}-\frac{3}{11}a^{8}-\frac{2}{11}a^{6}-\frac{1}{11}a^{5}+\frac{1}{11}a^{4}+\frac{3}{11}a^{3}+\frac{4}{11}a^{2}+\frac{4}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{38}-\frac{5}{11}a^{26}-\frac{4}{11}a^{25}-\frac{5}{11}a^{24}+\frac{1}{11}a^{23}+\frac{2}{11}a^{22}-\frac{1}{11}a^{21}+\frac{2}{11}a^{20}-\frac{1}{11}a^{19}-\frac{3}{11}a^{18}+\frac{3}{11}a^{17}+\frac{1}{11}a^{16}+\frac{3}{11}a^{15}-\frac{2}{11}a^{14}+\frac{5}{11}a^{13}+\frac{2}{11}a^{12}+\frac{3}{11}a^{11}+\frac{4}{11}a^{10}+\frac{4}{11}a^{9}-\frac{3}{11}a^{8}-\frac{1}{11}a^{7}-\frac{3}{11}a^{6}+\frac{2}{11}a^{5}-\frac{5}{11}a^{4}-\frac{4}{11}a^{3}-\frac{3}{11}a^{2}-\frac{3}{11}a+\frac{4}{11}$, $\frac{1}{5357}a^{39}-\frac{50}{5357}a^{38}+\frac{159}{5357}a^{37}+\frac{57}{5357}a^{36}+\frac{175}{5357}a^{35}+\frac{135}{5357}a^{34}+\frac{106}{5357}a^{33}-\frac{163}{5357}a^{32}+\frac{217}{5357}a^{31}+\frac{10}{487}a^{30}+\frac{112}{5357}a^{29}+\frac{221}{5357}a^{28}-\frac{118}{5357}a^{27}-\frac{400}{5357}a^{26}-\frac{1420}{5357}a^{25}-\frac{2213}{5357}a^{24}-\frac{2586}{5357}a^{23}+\frac{494}{5357}a^{22}+\frac{1123}{5357}a^{21}+\frac{167}{487}a^{20}+\frac{97}{5357}a^{19}-\frac{2469}{5357}a^{18}+\frac{393}{5357}a^{17}-\frac{1536}{5357}a^{16}-\frac{2526}{5357}a^{15}-\frac{1975}{5357}a^{14}-\frac{124}{5357}a^{13}+\frac{2490}{5357}a^{12}+\frac{1238}{5357}a^{11}+\frac{1532}{5357}a^{10}+\frac{2159}{5357}a^{9}+\frac{142}{5357}a^{8}-\frac{1365}{5357}a^{7}-\frac{821}{5357}a^{6}+\frac{226}{5357}a^{5}-\frac{2660}{5357}a^{4}+\frac{1021}{5357}a^{3}+\frac{996}{5357}a^{2}-\frac{1173}{5357}a+\frac{2311}{5357}$, $\frac{1}{5357}a^{40}+\frac{94}{5357}a^{38}+\frac{215}{5357}a^{37}+\frac{103}{5357}a^{36}+\frac{119}{5357}a^{35}+\frac{38}{5357}a^{34}-\frac{20}{487}a^{33}-\frac{141}{5357}a^{32}-\frac{241}{5357}a^{31}-\frac{232}{5357}a^{30}-\frac{23}{5357}a^{29}+\frac{218}{5357}a^{28}+\frac{31}{5357}a^{27}-\frac{1940}{5357}a^{26}-\frac{1137}{5357}a^{25}+\frac{2670}{5357}a^{24}-\frac{725}{5357}a^{23}+\frac{499}{5357}a^{22}-\frac{1427}{5357}a^{21}-\frac{2044}{5357}a^{20}+\frac{1894}{5357}a^{19}+\frac{2102}{5357}a^{18}+\frac{582}{5357}a^{17}+\frac{2003}{5357}a^{16}-\frac{239}{487}a^{15}+\frac{474}{5357}a^{14}+\frac{673}{5357}a^{13}-\frac{395}{5357}a^{12}+\frac{2557}{5357}a^{11}-\frac{2083}{5357}a^{10}-\frac{1970}{5357}a^{9}+\frac{865}{5357}a^{8}+\frac{2518}{5357}a^{7}+\frac{1058}{5357}a^{6}+\frac{2309}{5357}a^{5}-\frac{1950}{5357}a^{4}-\frac{1524}{5357}a^{3}-\frac{1047}{5357}a^{2}+\frac{640}{5357}a-\frac{843}{5357}$, $\frac{1}{5357}a^{41}+\frac{45}{5357}a^{38}-\frac{233}{5357}a^{37}+\frac{118}{5357}a^{36}+\frac{146}{5357}a^{35}+\frac{239}{5357}a^{34}+\frac{122}{5357}a^{33}-\frac{16}{5357}a^{32}-\frac{16}{487}a^{31}-\frac{136}{5357}a^{30}-\frac{83}{5357}a^{29}+\frac{18}{487}a^{28}-\frac{101}{5357}a^{27}-\frac{549}{5357}a^{26}-\frac{1184}{5357}a^{25}-\frac{2113}{5357}a^{24}-\frac{1865}{5357}a^{23}-\frac{624}{5357}a^{22}-\frac{953}{5357}a^{21}-\frac{1308}{5357}a^{20}+\frac{1750}{5357}a^{19}-\frac{1092}{5357}a^{18}+\frac{612}{5357}a^{17}-\frac{1423}{5357}a^{16}-\frac{2173}{5357}a^{15}+\frac{1750}{5357}a^{14}-\frac{1401}{5357}a^{13}-\frac{2613}{5357}a^{12}-\frac{1088}{5357}a^{11}-\frac{365}{5357}a^{10}+\frac{511}{5357}a^{9}+\frac{2319}{5357}a^{8}-\frac{1635}{5357}a^{7}+\frac{2537}{5357}a^{6}+\frac{2617}{5357}a^{5}+\frac{1120}{5357}a^{4}-\frac{2543}{5357}a^{3}-\frac{2402}{5357}a^{2}+\frac{331}{5357}a-\frac{32}{5357}$, $\frac{1}{35297273}a^{42}-\frac{178}{35297273}a^{41}+\frac{3248}{35297273}a^{40}+\frac{2773}{35297273}a^{39}+\frac{560496}{35297273}a^{38}+\frac{876594}{35297273}a^{37}+\frac{336231}{35297273}a^{36}-\frac{1160972}{35297273}a^{35}-\frac{1524040}{35297273}a^{34}-\frac{404268}{35297273}a^{33}+\frac{1043657}{35297273}a^{32}+\frac{36803}{3208843}a^{31}-\frac{896851}{35297273}a^{30}+\frac{894975}{35297273}a^{29}-\frac{527267}{35297273}a^{28}+\frac{1423280}{35297273}a^{27}+\frac{15947754}{35297273}a^{26}+\frac{12907434}{35297273}a^{25}+\frac{11650186}{35297273}a^{24}-\frac{9815380}{35297273}a^{23}-\frac{14567455}{35297273}a^{22}+\frac{11876003}{35297273}a^{21}+\frac{10282823}{35297273}a^{20}+\frac{9472315}{35297273}a^{19}-\frac{2818260}{35297273}a^{18}-\frac{16852426}{35297273}a^{17}+\frac{17332497}{35297273}a^{16}+\frac{8849870}{35297273}a^{15}+\frac{8130244}{35297273}a^{14}-\frac{2365549}{35297273}a^{13}-\frac{2978735}{35297273}a^{12}-\frac{6258720}{35297273}a^{11}-\frac{13591778}{35297273}a^{10}+\frac{6083219}{35297273}a^{9}+\frac{8244671}{35297273}a^{8}-\frac{17596710}{35297273}a^{7}+\frac{12198558}{35297273}a^{6}+\frac{17326196}{35297273}a^{5}+\frac{10605256}{35297273}a^{4}+\frac{11550007}{35297273}a^{3}-\frac{11487849}{35297273}a^{2}-\frac{6286856}{35297273}a+\frac{4688731}{35297273}$, $\frac{1}{35297273}a^{43}-\frac{2080}{35297273}a^{41}+\frac{1085}{35297273}a^{40}-\frac{150}{35297273}a^{39}+\frac{564561}{35297273}a^{38}-\frac{869933}{35297273}a^{37}-\frac{23368}{3208843}a^{36}+\frac{1043461}{35297273}a^{35}+\frac{244642}{35297273}a^{34}-\frac{545527}{35297273}a^{33}-\frac{259976}{35297273}a^{32}-\frac{43900}{35297273}a^{31}+\frac{623640}{35297273}a^{30}+\frac{510503}{35297273}a^{29}-\frac{572997}{35297273}a^{28}+\frac{75398}{3208843}a^{27}-\frac{9197785}{35297273}a^{26}+\frac{550398}{3208843}a^{25}-\frac{1793073}{35297273}a^{24}+\frac{16949332}{35297273}a^{23}+\frac{83407}{35297273}a^{22}-\frac{15843131}{35297273}a^{21}+\frac{17422600}{35297273}a^{20}-\frac{3754216}{35297273}a^{19}-\frac{679785}{35297273}a^{18}-\frac{8625495}{35297273}a^{17}-\frac{5661879}{35297273}a^{16}-\frac{7842985}{35297273}a^{15}+\frac{15578126}{35297273}a^{14}+\frac{11143815}{35297273}a^{13}+\frac{11737839}{35297273}a^{12}+\frac{5986923}{35297273}a^{11}+\frac{16064912}{35297273}a^{10}-\frac{10056493}{35297273}a^{9}+\frac{9474137}{35297273}a^{8}+\frac{791957}{3208843}a^{7}+\frac{7914779}{35297273}a^{6}-\frac{8243503}{35297273}a^{5}+\frac{984133}{3208843}a^{4}-\frac{12277063}{35297273}a^{3}-\frac{3130998}{35297273}a^{2}-\frac{12288908}{35297273}a-\frac{3539860}{35297273}$, $\frac{1}{11\!\cdots\!83}a^{44}+\frac{10\!\cdots\!59}{10\!\cdots\!53}a^{43}+\frac{48\!\cdots\!14}{11\!\cdots\!83}a^{42}+\frac{89\!\cdots\!14}{11\!\cdots\!83}a^{41}-\frac{29\!\cdots\!38}{11\!\cdots\!83}a^{40}-\frac{47\!\cdots\!29}{11\!\cdots\!83}a^{39}+\frac{80\!\cdots\!08}{11\!\cdots\!83}a^{38}-\frac{62\!\cdots\!05}{11\!\cdots\!83}a^{37}-\frac{39\!\cdots\!79}{10\!\cdots\!53}a^{36}-\frac{20\!\cdots\!99}{11\!\cdots\!83}a^{35}+\frac{51\!\cdots\!34}{11\!\cdots\!83}a^{34}+\frac{17\!\cdots\!92}{11\!\cdots\!83}a^{33}+\frac{39\!\cdots\!92}{11\!\cdots\!83}a^{32}-\frac{19\!\cdots\!73}{11\!\cdots\!83}a^{31}+\frac{67\!\cdots\!49}{11\!\cdots\!83}a^{30}-\frac{39\!\cdots\!36}{11\!\cdots\!83}a^{29}-\frac{35\!\cdots\!77}{11\!\cdots\!83}a^{28}+\frac{25\!\cdots\!20}{11\!\cdots\!83}a^{27}+\frac{56\!\cdots\!91}{11\!\cdots\!83}a^{26}+\frac{19\!\cdots\!24}{11\!\cdots\!83}a^{25}+\frac{10\!\cdots\!49}{10\!\cdots\!53}a^{24}-\frac{38\!\cdots\!45}{11\!\cdots\!83}a^{23}+\frac{53\!\cdots\!41}{11\!\cdots\!83}a^{22}+\frac{13\!\cdots\!70}{10\!\cdots\!53}a^{21}-\frac{57\!\cdots\!56}{11\!\cdots\!83}a^{20}+\frac{17\!\cdots\!70}{11\!\cdots\!83}a^{19}-\frac{11\!\cdots\!06}{11\!\cdots\!83}a^{18}-\frac{18\!\cdots\!15}{11\!\cdots\!83}a^{17}-\frac{33\!\cdots\!38}{11\!\cdots\!83}a^{16}-\frac{19\!\cdots\!38}{11\!\cdots\!83}a^{15}+\frac{88\!\cdots\!41}{11\!\cdots\!83}a^{14}-\frac{36\!\cdots\!39}{11\!\cdots\!83}a^{13}-\frac{42\!\cdots\!03}{11\!\cdots\!83}a^{12}-\frac{38\!\cdots\!84}{11\!\cdots\!83}a^{11}+\frac{84\!\cdots\!78}{96\!\cdots\!23}a^{10}-\frac{39\!\cdots\!41}{10\!\cdots\!53}a^{9}+\frac{41\!\cdots\!63}{11\!\cdots\!83}a^{8}-\frac{12\!\cdots\!57}{11\!\cdots\!83}a^{7}-\frac{42\!\cdots\!31}{11\!\cdots\!83}a^{6}+\frac{12\!\cdots\!84}{11\!\cdots\!83}a^{5}+\frac{49\!\cdots\!22}{11\!\cdots\!83}a^{4}-\frac{28\!\cdots\!34}{11\!\cdots\!83}a^{3}-\frac{17\!\cdots\!65}{11\!\cdots\!83}a^{2}+\frac{35\!\cdots\!47}{11\!\cdots\!83}a+\frac{20\!\cdots\!04}{11\!\cdots\!83}$ 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 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289)
 
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 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289, 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 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289);
 
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 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289);
 
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

\(\Q(\zeta_{9})^+\), 3.3.301401.1, 3.3.3721.1, 3.3.301401.2, 5.5.13845841.1, 9.9.27380039270784201.1, 15.15.9255142598391173348787179150721.1, 15.15.34438385608613556030837093619832841.2, 15.15.9876832533361318095112441.1, 15.15.34438385608613556030837093619832841.1

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 $15^{3}$ R $15^{3}$ $15^{3}$ ${\href{/padicField/11.3.0.1}{3} }^{15}$ ${\href{/padicField/13.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{15}$ $15^{3}$ ${\href{/padicField/37.5.0.1}{5} }^{9}$ $15^{3}$ $15^{3}$ ${\href{/padicField/47.3.0.1}{3} }^{15}$ ${\href{/padicField/53.5.0.1}{5} }^{9}$ $15^{3}$

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 3.15.20.65$x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$$3$$5$$20$$C_{15}$$[2]^{5}$
3.15.20.65$x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$$3$$5$$20$$C_{15}$$[2]^{5}$
3.15.20.65$x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$$3$$5$$20$$C_{15}$$[2]^{5}$
\(61\) Copy content Toggle raw display Deg $45$$15$$3$$42$