Properties

Label 45.45.235...625.1
Degree $45$
Signature $[45, 0]$
Discriminant $2.352\times 10^{112}$
Root discriminant \(314.15\)
Ramified primes $3,5,13$
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 - 15*x^44 - 90*x^43 + 2245*x^42 + 840*x^41 - 149682*x^40 + 233625*x^39 + 5893830*x^38 - 15309120*x^37 - 153159405*x^36 + 505833687*x^35 + 2780167860*x^34 - 10777640365*x^33 - 36390969195*x^32 + 160894854450*x^31 + 349558305696*x^30 - 1751940770010*x^29 - 2483116182570*x^28 + 14226276458010*x^27 + 13037929556940*x^26 - 87229174471284*x^25 - 50137706272620*x^24 + 406336593221610*x^23 + 138267603359730*x^22 - 1439822630956295*x^21 - 262195938148443*x^20 + 3868718864967360*x^19 + 311910060390275*x^18 - 7821266721741750*x^17 - 175081417856730*x^16 + 11738415157749574*x^15 - 32411856380685*x^14 - 12806776910595090*x^13 + 44301971232950*x^12 + 9836514943009755*x^11 + 117791800035840*x^10 - 5063789878500180*x^9 - 176503490451285*x^8 + 1618208036238660*x^7 + 90536277582715*x^6 - 283553373081756*x^5 - 18640864878495*x^4 + 21751317924475*x^3 + 974579665770*x^2 - 434206226325*x + 15265738099)
 
gp: K = bnfinit(y^45 - 15*y^44 - 90*y^43 + 2245*y^42 + 840*y^41 - 149682*y^40 + 233625*y^39 + 5893830*y^38 - 15309120*y^37 - 153159405*y^36 + 505833687*y^35 + 2780167860*y^34 - 10777640365*y^33 - 36390969195*y^32 + 160894854450*y^31 + 349558305696*y^30 - 1751940770010*y^29 - 2483116182570*y^28 + 14226276458010*y^27 + 13037929556940*y^26 - 87229174471284*y^25 - 50137706272620*y^24 + 406336593221610*y^23 + 138267603359730*y^22 - 1439822630956295*y^21 - 262195938148443*y^20 + 3868718864967360*y^19 + 311910060390275*y^18 - 7821266721741750*y^17 - 175081417856730*y^16 + 11738415157749574*y^15 - 32411856380685*y^14 - 12806776910595090*y^13 + 44301971232950*y^12 + 9836514943009755*y^11 + 117791800035840*y^10 - 5063789878500180*y^9 - 176503490451285*y^8 + 1618208036238660*y^7 + 90536277582715*y^6 - 283553373081756*y^5 - 18640864878495*y^4 + 21751317924475*y^3 + 974579665770*y^2 - 434206226325*y + 15265738099, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 15*x^44 - 90*x^43 + 2245*x^42 + 840*x^41 - 149682*x^40 + 233625*x^39 + 5893830*x^38 - 15309120*x^37 - 153159405*x^36 + 505833687*x^35 + 2780167860*x^34 - 10777640365*x^33 - 36390969195*x^32 + 160894854450*x^31 + 349558305696*x^30 - 1751940770010*x^29 - 2483116182570*x^28 + 14226276458010*x^27 + 13037929556940*x^26 - 87229174471284*x^25 - 50137706272620*x^24 + 406336593221610*x^23 + 138267603359730*x^22 - 1439822630956295*x^21 - 262195938148443*x^20 + 3868718864967360*x^19 + 311910060390275*x^18 - 7821266721741750*x^17 - 175081417856730*x^16 + 11738415157749574*x^15 - 32411856380685*x^14 - 12806776910595090*x^13 + 44301971232950*x^12 + 9836514943009755*x^11 + 117791800035840*x^10 - 5063789878500180*x^9 - 176503490451285*x^8 + 1618208036238660*x^7 + 90536277582715*x^6 - 283553373081756*x^5 - 18640864878495*x^4 + 21751317924475*x^3 + 974579665770*x^2 - 434206226325*x + 15265738099);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 15*x^44 - 90*x^43 + 2245*x^42 + 840*x^41 - 149682*x^40 + 233625*x^39 + 5893830*x^38 - 15309120*x^37 - 153159405*x^36 + 505833687*x^35 + 2780167860*x^34 - 10777640365*x^33 - 36390969195*x^32 + 160894854450*x^31 + 349558305696*x^30 - 1751940770010*x^29 - 2483116182570*x^28 + 14226276458010*x^27 + 13037929556940*x^26 - 87229174471284*x^25 - 50137706272620*x^24 + 406336593221610*x^23 + 138267603359730*x^22 - 1439822630956295*x^21 - 262195938148443*x^20 + 3868718864967360*x^19 + 311910060390275*x^18 - 7821266721741750*x^17 - 175081417856730*x^16 + 11738415157749574*x^15 - 32411856380685*x^14 - 12806776910595090*x^13 + 44301971232950*x^12 + 9836514943009755*x^11 + 117791800035840*x^10 - 5063789878500180*x^9 - 176503490451285*x^8 + 1618208036238660*x^7 + 90536277582715*x^6 - 283553373081756*x^5 - 18640864878495*x^4 + 21751317924475*x^3 + 974579665770*x^2 - 434206226325*x + 15265738099)
 

\( x^{45} - 15 x^{44} - 90 x^{43} + 2245 x^{42} + 840 x^{41} - 149682 x^{40} + 233625 x^{39} + 5893830 x^{38} - 15309120 x^{37} - 153159405 x^{36} + 505833687 x^{35} + \cdots + 15265738099 \) 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:   \(235\!\cdots\!625\) \(\medspace = 3^{60}\cdot 5^{72}\cdot 13^{30}\) 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:  \(314.15\)
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:   \(3\), \(5\), \(13\) 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:  \(2925=3^{2}\cdot 5^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2925}(256,·)$, $\chi_{2925}(1,·)$, $\chi_{2925}(646,·)$, $\chi_{2925}(391,·)$, $\chi_{2925}(1036,·)$, $\chi_{2925}(781,·)$, $\chi_{2925}(16,·)$, $\chi_{2925}(1426,·)$, $\chi_{2925}(1171,·)$, $\chi_{2925}(406,·)$, $\chi_{2925}(1816,·)$, $\chi_{2925}(1561,·)$, $\chi_{2925}(796,·)$, $\chi_{2925}(2206,·)$, $\chi_{2925}(1951,·)$, $\chi_{2925}(1186,·)$, $\chi_{2925}(2596,·)$, $\chi_{2925}(2341,·)$, $\chi_{2925}(1576,·)$, $\chi_{2925}(2731,·)$, $\chi_{2925}(1966,·)$, $\chi_{2925}(2356,·)$, $\chi_{2925}(2746,·)$, $\chi_{2925}(61,·)$, $\chi_{2925}(451,·)$, $\chi_{2925}(196,·)$, $\chi_{2925}(841,·)$, $\chi_{2925}(586,·)$, $\chi_{2925}(1231,·)$, $\chi_{2925}(976,·)$, $\chi_{2925}(211,·)$, $\chi_{2925}(1621,·)$, $\chi_{2925}(1366,·)$, $\chi_{2925}(601,·)$, $\chi_{2925}(2011,·)$, $\chi_{2925}(1756,·)$, $\chi_{2925}(991,·)$, $\chi_{2925}(2401,·)$, $\chi_{2925}(2146,·)$, $\chi_{2925}(1381,·)$, $\chi_{2925}(2791,·)$, $\chi_{2925}(2536,·)$, $\chi_{2925}(1771,·)$, $\chi_{2925}(2161,·)$, $\chi_{2925}(2551,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{17}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{18}-\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{19}-\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{19}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{18}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{35}-\frac{1}{4}a^{34}-\frac{1}{4}a^{33}-\frac{1}{4}a^{31}-\frac{1}{4}a^{29}-\frac{1}{4}a^{28}-\frac{1}{4}a^{27}-\frac{1}{4}a^{26}-\frac{1}{4}a^{25}-\frac{1}{4}a^{23}-\frac{1}{4}a^{19}-\frac{1}{2}a^{18}-\frac{1}{4}a^{16}-\frac{1}{4}a^{14}+\frac{1}{4}a^{12}-\frac{1}{2}a^{11}+\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{28}a^{36}+\frac{3}{28}a^{35}+\frac{3}{28}a^{34}-\frac{1}{14}a^{33}-\frac{5}{28}a^{32}-\frac{1}{7}a^{31}+\frac{1}{28}a^{30}-\frac{5}{28}a^{29}-\frac{1}{28}a^{28}-\frac{1}{28}a^{27}+\frac{3}{28}a^{26}-\frac{1}{7}a^{25}+\frac{3}{28}a^{24}+\frac{1}{14}a^{23}-\frac{1}{7}a^{22}-\frac{1}{28}a^{20}-\frac{1}{14}a^{19}+\frac{1}{14}a^{18}+\frac{5}{28}a^{17}-\frac{1}{7}a^{16}+\frac{13}{28}a^{15}+\frac{11}{28}a^{13}+\frac{1}{14}a^{12}-\frac{3}{28}a^{11}+\frac{3}{28}a^{10}-\frac{5}{14}a^{9}+\frac{1}{28}a^{8}-\frac{9}{28}a^{7}-\frac{5}{28}a^{6}-\frac{9}{28}a^{5}+\frac{2}{7}a^{4}+\frac{3}{28}a^{3}+\frac{5}{28}a^{2}-\frac{9}{28}a+\frac{2}{7}$, $\frac{1}{28}a^{37}+\frac{1}{28}a^{35}-\frac{1}{7}a^{34}-\frac{3}{14}a^{33}-\frac{3}{28}a^{32}+\frac{3}{14}a^{31}+\frac{3}{14}a^{30}-\frac{1}{4}a^{29}-\frac{5}{28}a^{28}-\frac{1}{28}a^{27}-\frac{3}{14}a^{26}-\frac{3}{14}a^{25}-\frac{1}{4}a^{24}-\frac{3}{28}a^{23}-\frac{1}{14}a^{22}-\frac{1}{28}a^{21}+\frac{1}{28}a^{20}-\frac{13}{28}a^{19}-\frac{1}{28}a^{18}-\frac{5}{28}a^{17}+\frac{1}{7}a^{16}+\frac{3}{28}a^{15}-\frac{5}{14}a^{14}+\frac{11}{28}a^{13}-\frac{1}{14}a^{12}+\frac{3}{7}a^{11}+\frac{1}{14}a^{10}-\frac{1}{7}a^{9}+\frac{1}{14}a^{8}-\frac{13}{28}a^{7}+\frac{13}{28}a^{6}-\frac{1}{7}a^{3}-\frac{3}{28}a^{2}-\frac{1}{2}a-\frac{3}{28}$, $\frac{1}{28}a^{38}-\frac{1}{14}a^{34}+\frac{3}{14}a^{33}-\frac{3}{28}a^{32}+\frac{3}{28}a^{31}+\frac{3}{14}a^{30}-\frac{1}{4}a^{29}-\frac{1}{4}a^{28}+\frac{1}{14}a^{27}-\frac{1}{14}a^{26}+\frac{1}{7}a^{25}-\frac{3}{14}a^{24}+\frac{3}{28}a^{23}+\frac{3}{28}a^{22}+\frac{1}{28}a^{21}+\frac{1}{14}a^{20}+\frac{2}{7}a^{19}-\frac{1}{4}a^{18}+\frac{13}{28}a^{17}-\frac{1}{2}a^{16}+\frac{5}{28}a^{15}+\frac{1}{7}a^{14}-\frac{13}{28}a^{13}+\frac{3}{28}a^{12}-\frac{9}{28}a^{11}-\frac{1}{2}a^{10}+\frac{5}{28}a^{9}-\frac{1}{2}a^{8}+\frac{1}{28}a^{7}+\frac{3}{7}a^{6}-\frac{3}{7}a^{5}+\frac{9}{28}a^{4}-\frac{3}{14}a^{3}-\frac{3}{7}a^{2}-\frac{1}{28}a+\frac{13}{28}$, $\frac{1}{28}a^{39}-\frac{1}{14}a^{35}+\frac{3}{14}a^{34}-\frac{3}{28}a^{33}+\frac{3}{28}a^{32}+\frac{3}{14}a^{31}-\frac{1}{4}a^{30}-\frac{1}{4}a^{29}+\frac{1}{14}a^{28}-\frac{1}{14}a^{27}+\frac{1}{7}a^{26}-\frac{3}{14}a^{25}+\frac{3}{28}a^{24}+\frac{3}{28}a^{23}+\frac{1}{28}a^{22}+\frac{1}{14}a^{21}-\frac{3}{14}a^{20}-\frac{1}{4}a^{19}-\frac{1}{28}a^{18}-\frac{9}{28}a^{16}+\frac{1}{7}a^{15}-\frac{13}{28}a^{14}+\frac{3}{28}a^{13}-\frac{9}{28}a^{12}-\frac{1}{2}a^{11}+\frac{5}{28}a^{10}-\frac{1}{2}a^{9}-\frac{13}{28}a^{8}+\frac{3}{7}a^{7}+\frac{1}{14}a^{6}+\frac{9}{28}a^{5}+\frac{2}{7}a^{4}+\frac{1}{14}a^{3}+\frac{13}{28}a^{2}-\frac{1}{28}a-\frac{1}{2}$, $\frac{1}{96656}a^{40}-\frac{169}{24164}a^{39}-\frac{15}{48328}a^{38}+\frac{27}{3452}a^{37}-\frac{161}{13808}a^{36}+\frac{2987}{24164}a^{35}+\frac{1663}{96656}a^{34}+\frac{7101}{48328}a^{33}-\frac{23089}{96656}a^{32}+\frac{226}{6041}a^{31}+\frac{5041}{24164}a^{30}-\frac{2459}{24164}a^{29}-\frac{363}{6041}a^{28}-\frac{149}{24164}a^{27}-\frac{4885}{48328}a^{26}+\frac{9501}{48328}a^{25}+\frac{3833}{24164}a^{24}-\frac{2147}{24164}a^{23}-\frac{7155}{48328}a^{22}+\frac{4535}{48328}a^{21}+\frac{1229}{12082}a^{20}+\frac{1801}{6904}a^{19}+\frac{2437}{12082}a^{18}-\frac{10019}{48328}a^{17}-\frac{17181}{96656}a^{16}-\frac{3911}{24164}a^{15}+\frac{8823}{24164}a^{14}+\frac{11975}{24164}a^{13}+\frac{13271}{96656}a^{12}-\frac{3747}{48328}a^{11}-\frac{677}{1726}a^{10}-\frac{8607}{48328}a^{9}-\frac{10231}{96656}a^{8}+\frac{21807}{48328}a^{7}+\frac{29857}{96656}a^{6}-\frac{19681}{48328}a^{5}-\frac{24441}{96656}a^{4}+\frac{725}{24164}a^{3}+\frac{12749}{96656}a^{2}+\frac{3889}{24164}a-\frac{927}{96656}$, $\frac{1}{96656}a^{41}-\frac{671}{48328}a^{39}+\frac{297}{24164}a^{38}-\frac{967}{96656}a^{37}-\frac{205}{24164}a^{36}+\frac{11187}{96656}a^{35}+\frac{1341}{48328}a^{34}-\frac{5289}{96656}a^{33}+\frac{1969}{12082}a^{32}+\frac{5141}{24164}a^{31}+\frac{1475}{6041}a^{30}-\frac{2461}{24164}a^{29}-\frac{235}{12082}a^{28}-\frac{11295}{48328}a^{27}-\frac{11633}{48328}a^{26}-\frac{3819}{24164}a^{25}+\frac{3001}{12082}a^{24}-\frac{3315}{48328}a^{23}-\frac{2897}{48328}a^{22}-\frac{213}{6041}a^{21}+\frac{6369}{48328}a^{20}-\frac{158}{6041}a^{19}+\frac{10473}{48328}a^{18}-\frac{34481}{96656}a^{17}+\frac{634}{6041}a^{16}+\frac{3315}{12082}a^{15}+\frac{6089}{24164}a^{14}-\frac{37869}{96656}a^{13}-\frac{3287}{6904}a^{12}+\frac{2139}{24164}a^{11}-\frac{3861}{48328}a^{10}-\frac{27463}{96656}a^{9}+\frac{14003}{48328}a^{8}-\frac{39651}{96656}a^{7}-\frac{2677}{48328}a^{6}-\frac{42397}{96656}a^{5}+\frac{1423}{6041}a^{4}+\frac{43481}{96656}a^{3}-\frac{8523}{24164}a^{2}+\frac{1499}{13808}a-\frac{8227}{24164}$, $\frac{1}{20\!\cdots\!44}a^{42}-\frac{63\!\cdots\!39}{20\!\cdots\!44}a^{41}+\frac{41\!\cdots\!59}{10\!\cdots\!72}a^{40}+\frac{20\!\cdots\!67}{14\!\cdots\!96}a^{39}-\frac{20\!\cdots\!67}{20\!\cdots\!44}a^{38}+\frac{22\!\cdots\!13}{20\!\cdots\!44}a^{37}+\frac{12\!\cdots\!47}{20\!\cdots\!44}a^{36}+\frac{78\!\cdots\!05}{20\!\cdots\!44}a^{35}+\frac{50\!\cdots\!09}{20\!\cdots\!44}a^{34}-\frac{28\!\cdots\!93}{20\!\cdots\!44}a^{33}+\frac{21\!\cdots\!22}{12\!\cdots\!59}a^{32}-\frac{29\!\cdots\!27}{12\!\cdots\!59}a^{31}-\frac{99\!\cdots\!93}{50\!\cdots\!36}a^{30}-\frac{26\!\cdots\!24}{12\!\cdots\!59}a^{29}-\frac{12\!\cdots\!89}{14\!\cdots\!96}a^{28}+\frac{50\!\cdots\!77}{25\!\cdots\!18}a^{27}-\frac{38\!\cdots\!35}{10\!\cdots\!72}a^{26}-\frac{74\!\cdots\!47}{12\!\cdots\!59}a^{25}-\frac{74\!\cdots\!67}{10\!\cdots\!72}a^{24}-\frac{64\!\cdots\!93}{25\!\cdots\!18}a^{23}+\frac{29\!\cdots\!63}{14\!\cdots\!96}a^{22}-\frac{20\!\cdots\!47}{10\!\cdots\!72}a^{21}+\frac{12\!\cdots\!63}{10\!\cdots\!72}a^{20}+\frac{65\!\cdots\!39}{14\!\cdots\!96}a^{19}-\frac{60\!\cdots\!31}{20\!\cdots\!44}a^{18}-\frac{89\!\cdots\!93}{20\!\cdots\!44}a^{17}-\frac{11\!\cdots\!15}{50\!\cdots\!36}a^{16}-\frac{12\!\cdots\!51}{25\!\cdots\!18}a^{15}+\frac{40\!\cdots\!71}{20\!\cdots\!44}a^{14}-\frac{88\!\cdots\!75}{20\!\cdots\!44}a^{13}+\frac{30\!\cdots\!43}{10\!\cdots\!72}a^{12}-\frac{90\!\cdots\!35}{14\!\cdots\!96}a^{11}+\frac{12\!\cdots\!57}{29\!\cdots\!92}a^{10}+\frac{61\!\cdots\!07}{20\!\cdots\!44}a^{9}+\frac{71\!\cdots\!97}{29\!\cdots\!92}a^{8}+\frac{17\!\cdots\!67}{20\!\cdots\!44}a^{7}+\frac{24\!\cdots\!93}{20\!\cdots\!44}a^{6}+\frac{22\!\cdots\!11}{20\!\cdots\!44}a^{5}-\frac{67\!\cdots\!03}{20\!\cdots\!44}a^{4}-\frac{11\!\cdots\!79}{20\!\cdots\!44}a^{3}+\frac{91\!\cdots\!21}{20\!\cdots\!44}a^{2}-\frac{46\!\cdots\!19}{20\!\cdots\!44}a-\frac{18\!\cdots\!01}{50\!\cdots\!36}$, $\frac{1}{62\!\cdots\!08}a^{43}-\frac{33}{62\!\cdots\!08}a^{42}-\frac{30\!\cdots\!85}{62\!\cdots\!08}a^{41}+\frac{18\!\cdots\!89}{15\!\cdots\!52}a^{40}-\frac{27\!\cdots\!33}{62\!\cdots\!08}a^{39}+\frac{23\!\cdots\!55}{62\!\cdots\!08}a^{38}-\frac{21\!\cdots\!81}{15\!\cdots\!52}a^{37}-\frac{30\!\cdots\!67}{62\!\cdots\!08}a^{36}-\frac{10\!\cdots\!01}{15\!\cdots\!52}a^{35}+\frac{16\!\cdots\!41}{62\!\cdots\!08}a^{34}+\frac{36\!\cdots\!47}{62\!\cdots\!08}a^{33}+\frac{68\!\cdots\!65}{31\!\cdots\!04}a^{32}+\frac{53\!\cdots\!77}{38\!\cdots\!13}a^{31}+\frac{12\!\cdots\!80}{38\!\cdots\!13}a^{30}-\frac{74\!\cdots\!73}{31\!\cdots\!04}a^{29}-\frac{88\!\cdots\!19}{77\!\cdots\!26}a^{28}-\frac{91\!\cdots\!09}{15\!\cdots\!52}a^{27}+\frac{76\!\cdots\!99}{31\!\cdots\!04}a^{26}-\frac{40\!\cdots\!83}{31\!\cdots\!04}a^{25}-\frac{24\!\cdots\!47}{15\!\cdots\!52}a^{24}+\frac{93\!\cdots\!43}{15\!\cdots\!52}a^{23}-\frac{92\!\cdots\!58}{38\!\cdots\!13}a^{22}-\frac{66\!\cdots\!13}{31\!\cdots\!04}a^{21}+\frac{18\!\cdots\!53}{77\!\cdots\!26}a^{20}-\frac{20\!\cdots\!47}{62\!\cdots\!08}a^{19}-\frac{24\!\cdots\!29}{62\!\cdots\!08}a^{18}+\frac{50\!\cdots\!03}{62\!\cdots\!08}a^{17}+\frac{31\!\cdots\!51}{31\!\cdots\!04}a^{16}-\frac{29\!\cdots\!03}{89\!\cdots\!44}a^{15}-\frac{49\!\cdots\!01}{62\!\cdots\!08}a^{14}-\frac{14\!\cdots\!81}{89\!\cdots\!44}a^{13}+\frac{12\!\cdots\!61}{31\!\cdots\!04}a^{12}+\frac{27\!\cdots\!19}{62\!\cdots\!08}a^{11}+\frac{38\!\cdots\!11}{62\!\cdots\!08}a^{10}-\frac{76\!\cdots\!77}{15\!\cdots\!52}a^{9}-\frac{22\!\cdots\!47}{62\!\cdots\!08}a^{8}+\frac{74\!\cdots\!17}{31\!\cdots\!04}a^{7}+\frac{12\!\cdots\!81}{62\!\cdots\!08}a^{6}-\frac{18\!\cdots\!57}{15\!\cdots\!52}a^{5}-\frac{19\!\cdots\!49}{89\!\cdots\!44}a^{4}+\frac{19\!\cdots\!15}{44\!\cdots\!72}a^{3}-\frac{15\!\cdots\!51}{62\!\cdots\!08}a^{2}-\frac{35\!\cdots\!59}{62\!\cdots\!08}a+\frac{13\!\cdots\!69}{31\!\cdots\!04}$, $\frac{1}{29\!\cdots\!92}a^{44}+\frac{14\!\cdots\!31}{29\!\cdots\!92}a^{43}-\frac{18\!\cdots\!97}{29\!\cdots\!92}a^{42}+\frac{84\!\cdots\!41}{29\!\cdots\!92}a^{41}+\frac{52\!\cdots\!59}{29\!\cdots\!92}a^{40}-\frac{32\!\cdots\!03}{29\!\cdots\!92}a^{39}-\frac{62\!\cdots\!43}{52\!\cdots\!82}a^{38}+\frac{28\!\cdots\!15}{20\!\cdots\!28}a^{37}+\frac{26\!\cdots\!07}{36\!\cdots\!74}a^{36}+\frac{25\!\cdots\!75}{36\!\cdots\!74}a^{35}-\frac{53\!\cdots\!97}{41\!\cdots\!56}a^{34}-\frac{35\!\cdots\!27}{29\!\cdots\!92}a^{33}-\frac{56\!\cdots\!83}{10\!\cdots\!64}a^{32}+\frac{84\!\cdots\!39}{36\!\cdots\!74}a^{31}+\frac{35\!\cdots\!21}{20\!\cdots\!28}a^{30}-\frac{12\!\cdots\!54}{18\!\cdots\!37}a^{29}-\frac{17\!\cdots\!31}{72\!\cdots\!48}a^{28}+\frac{41\!\cdots\!52}{18\!\cdots\!37}a^{27}+\frac{43\!\cdots\!92}{18\!\cdots\!37}a^{26}-\frac{14\!\cdots\!53}{72\!\cdots\!48}a^{25}-\frac{86\!\cdots\!19}{10\!\cdots\!64}a^{24}-\frac{23\!\cdots\!41}{20\!\cdots\!28}a^{23}+\frac{51\!\cdots\!95}{18\!\cdots\!37}a^{22}+\frac{25\!\cdots\!06}{18\!\cdots\!37}a^{21}-\frac{23\!\cdots\!93}{29\!\cdots\!92}a^{20}-\frac{48\!\cdots\!29}{29\!\cdots\!92}a^{19}-\frac{12\!\cdots\!63}{29\!\cdots\!92}a^{18}+\frac{83\!\cdots\!49}{29\!\cdots\!92}a^{17}+\frac{40\!\cdots\!63}{29\!\cdots\!92}a^{16}+\frac{80\!\cdots\!51}{29\!\cdots\!92}a^{15}-\frac{97\!\cdots\!15}{29\!\cdots\!92}a^{14}+\frac{14\!\cdots\!81}{29\!\cdots\!92}a^{13}+\frac{71\!\cdots\!73}{29\!\cdots\!92}a^{12}+\frac{11\!\cdots\!07}{29\!\cdots\!92}a^{11}-\frac{15\!\cdots\!25}{14\!\cdots\!96}a^{10}+\frac{86\!\cdots\!77}{20\!\cdots\!28}a^{9}-\frac{11\!\cdots\!41}{72\!\cdots\!48}a^{8}+\frac{54\!\cdots\!97}{14\!\cdots\!96}a^{7}+\frac{97\!\cdots\!15}{14\!\cdots\!96}a^{6}-\frac{30\!\cdots\!05}{72\!\cdots\!48}a^{5}-\frac{40\!\cdots\!19}{14\!\cdots\!96}a^{4}+\frac{19\!\cdots\!11}{14\!\cdots\!96}a^{3}+\frac{11\!\cdots\!97}{29\!\cdots\!92}a^{2}-\frac{64\!\cdots\!65}{29\!\cdots\!92}a-\frac{25\!\cdots\!01}{72\!\cdots\!48}$ 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 - 15*x^44 - 90*x^43 + 2245*x^42 + 840*x^41 - 149682*x^40 + 233625*x^39 + 5893830*x^38 - 15309120*x^37 - 153159405*x^36 + 505833687*x^35 + 2780167860*x^34 - 10777640365*x^33 - 36390969195*x^32 + 160894854450*x^31 + 349558305696*x^30 - 1751940770010*x^29 - 2483116182570*x^28 + 14226276458010*x^27 + 13037929556940*x^26 - 87229174471284*x^25 - 50137706272620*x^24 + 406336593221610*x^23 + 138267603359730*x^22 - 1439822630956295*x^21 - 262195938148443*x^20 + 3868718864967360*x^19 + 311910060390275*x^18 - 7821266721741750*x^17 - 175081417856730*x^16 + 11738415157749574*x^15 - 32411856380685*x^14 - 12806776910595090*x^13 + 44301971232950*x^12 + 9836514943009755*x^11 + 117791800035840*x^10 - 5063789878500180*x^9 - 176503490451285*x^8 + 1618208036238660*x^7 + 90536277582715*x^6 - 283553373081756*x^5 - 18640864878495*x^4 + 21751317924475*x^3 + 974579665770*x^2 - 434206226325*x + 15265738099)
 
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 - 15*x^44 - 90*x^43 + 2245*x^42 + 840*x^41 - 149682*x^40 + 233625*x^39 + 5893830*x^38 - 15309120*x^37 - 153159405*x^36 + 505833687*x^35 + 2780167860*x^34 - 10777640365*x^33 - 36390969195*x^32 + 160894854450*x^31 + 349558305696*x^30 - 1751940770010*x^29 - 2483116182570*x^28 + 14226276458010*x^27 + 13037929556940*x^26 - 87229174471284*x^25 - 50137706272620*x^24 + 406336593221610*x^23 + 138267603359730*x^22 - 1439822630956295*x^21 - 262195938148443*x^20 + 3868718864967360*x^19 + 311910060390275*x^18 - 7821266721741750*x^17 - 175081417856730*x^16 + 11738415157749574*x^15 - 32411856380685*x^14 - 12806776910595090*x^13 + 44301971232950*x^12 + 9836514943009755*x^11 + 117791800035840*x^10 - 5063789878500180*x^9 - 176503490451285*x^8 + 1618208036238660*x^7 + 90536277582715*x^6 - 283553373081756*x^5 - 18640864878495*x^4 + 21751317924475*x^3 + 974579665770*x^2 - 434206226325*x + 15265738099, 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 - 15*x^44 - 90*x^43 + 2245*x^42 + 840*x^41 - 149682*x^40 + 233625*x^39 + 5893830*x^38 - 15309120*x^37 - 153159405*x^36 + 505833687*x^35 + 2780167860*x^34 - 10777640365*x^33 - 36390969195*x^32 + 160894854450*x^31 + 349558305696*x^30 - 1751940770010*x^29 - 2483116182570*x^28 + 14226276458010*x^27 + 13037929556940*x^26 - 87229174471284*x^25 - 50137706272620*x^24 + 406336593221610*x^23 + 138267603359730*x^22 - 1439822630956295*x^21 - 262195938148443*x^20 + 3868718864967360*x^19 + 311910060390275*x^18 - 7821266721741750*x^17 - 175081417856730*x^16 + 11738415157749574*x^15 - 32411856380685*x^14 - 12806776910595090*x^13 + 44301971232950*x^12 + 9836514943009755*x^11 + 117791800035840*x^10 - 5063789878500180*x^9 - 176503490451285*x^8 + 1618208036238660*x^7 + 90536277582715*x^6 - 283553373081756*x^5 - 18640864878495*x^4 + 21751317924475*x^3 + 974579665770*x^2 - 434206226325*x + 15265738099);
 
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 - 15*x^44 - 90*x^43 + 2245*x^42 + 840*x^41 - 149682*x^40 + 233625*x^39 + 5893830*x^38 - 15309120*x^37 - 153159405*x^36 + 505833687*x^35 + 2780167860*x^34 - 10777640365*x^33 - 36390969195*x^32 + 160894854450*x^31 + 349558305696*x^30 - 1751940770010*x^29 - 2483116182570*x^28 + 14226276458010*x^27 + 13037929556940*x^26 - 87229174471284*x^25 - 50137706272620*x^24 + 406336593221610*x^23 + 138267603359730*x^22 - 1439822630956295*x^21 - 262195938148443*x^20 + 3868718864967360*x^19 + 311910060390275*x^18 - 7821266721741750*x^17 - 175081417856730*x^16 + 11738415157749574*x^15 - 32411856380685*x^14 - 12806776910595090*x^13 + 44301971232950*x^12 + 9836514943009755*x^11 + 117791800035840*x^10 - 5063789878500180*x^9 - 176503490451285*x^8 + 1618208036238660*x^7 + 90536277582715*x^6 - 283553373081756*x^5 - 18640864878495*x^4 + 21751317924475*x^3 + 974579665770*x^2 - 434206226325*x + 15265738099);
 
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

3.3.13689.2, \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.169.1, 5.5.390625.1, 9.9.2565164201769.1, 15.15.28650929863719871488153934478759765625.1, 15.15.207828545629978179931640625.1, 15.15.28650929863719871488153934478759765625.2, 15.15.8217006435930728912353515625.1

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 R ${\href{/padicField/7.3.0.1}{3} }^{15}$ $15^{3}$ R $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{15}$ $15^{3}$ ${\href{/padicField/53.5.0.1}{5} }^{9}$ $15^{3}$

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 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}$
\(13\) Copy content Toggle raw display Deg $45$$3$$15$$30$