Properties

Label 45.45.148...969.1
Degree $45$
Signature $[45, 0]$
Discriminant $1.487\times 10^{106}$
Root discriminant \(228.76\)
Ramified primes $3,31$
Class number not computed
Class group not computed
Galois group $C_{45}$ (as 45T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 9*x^44 - 117*x^43 + 1356*x^42 + 4455*x^41 - 86382*x^40 + 4596*x^39 + 3024342*x^38 - 5886234*x^37 - 62776190*x^36 + 223064037*x^35 + 751420521*x^34 - 4412144961*x^33 - 3725440704*x^32 + 54057041307*x^31 - 28786222110*x^30 - 426126343077*x^29 + 673008341841*x^28 + 2086637912508*x^27 - 5873721786378*x^26 - 5053254569676*x^25 + 30385493847345*x^24 - 5904503198856*x^23 - 98922911866245*x^22 + 93605350409913*x^21 + 191046717571311*x^20 - 349809055484526*x^19 - 147155959283342*x^18 + 707251464652203*x^17 - 217826629489476*x^16 - 797405893685064*x^15 + 705978096992721*x^14 + 378713501361567*x^13 - 782478874269585*x^12 + 139525392055671*x^11 + 385252617992454*x^10 - 250562776470715*x^9 - 36607492213488*x^8 + 95351499735039*x^7 - 31186566059442*x^6 - 5609742817023*x^5 + 6576945066567*x^4 - 1857401457582*x^3 + 224711418717*x^2 - 8430571116*x - 155181097)
 
gp: K = bnfinit(y^45 - 9*y^44 - 117*y^43 + 1356*y^42 + 4455*y^41 - 86382*y^40 + 4596*y^39 + 3024342*y^38 - 5886234*y^37 - 62776190*y^36 + 223064037*y^35 + 751420521*y^34 - 4412144961*y^33 - 3725440704*y^32 + 54057041307*y^31 - 28786222110*y^30 - 426126343077*y^29 + 673008341841*y^28 + 2086637912508*y^27 - 5873721786378*y^26 - 5053254569676*y^25 + 30385493847345*y^24 - 5904503198856*y^23 - 98922911866245*y^22 + 93605350409913*y^21 + 191046717571311*y^20 - 349809055484526*y^19 - 147155959283342*y^18 + 707251464652203*y^17 - 217826629489476*y^16 - 797405893685064*y^15 + 705978096992721*y^14 + 378713501361567*y^13 - 782478874269585*y^12 + 139525392055671*y^11 + 385252617992454*y^10 - 250562776470715*y^9 - 36607492213488*y^8 + 95351499735039*y^7 - 31186566059442*y^6 - 5609742817023*y^5 + 6576945066567*y^4 - 1857401457582*y^3 + 224711418717*y^2 - 8430571116*y - 155181097, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 9*x^44 - 117*x^43 + 1356*x^42 + 4455*x^41 - 86382*x^40 + 4596*x^39 + 3024342*x^38 - 5886234*x^37 - 62776190*x^36 + 223064037*x^35 + 751420521*x^34 - 4412144961*x^33 - 3725440704*x^32 + 54057041307*x^31 - 28786222110*x^30 - 426126343077*x^29 + 673008341841*x^28 + 2086637912508*x^27 - 5873721786378*x^26 - 5053254569676*x^25 + 30385493847345*x^24 - 5904503198856*x^23 - 98922911866245*x^22 + 93605350409913*x^21 + 191046717571311*x^20 - 349809055484526*x^19 - 147155959283342*x^18 + 707251464652203*x^17 - 217826629489476*x^16 - 797405893685064*x^15 + 705978096992721*x^14 + 378713501361567*x^13 - 782478874269585*x^12 + 139525392055671*x^11 + 385252617992454*x^10 - 250562776470715*x^9 - 36607492213488*x^8 + 95351499735039*x^7 - 31186566059442*x^6 - 5609742817023*x^5 + 6576945066567*x^4 - 1857401457582*x^3 + 224711418717*x^2 - 8430571116*x - 155181097);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 9*x^44 - 117*x^43 + 1356*x^42 + 4455*x^41 - 86382*x^40 + 4596*x^39 + 3024342*x^38 - 5886234*x^37 - 62776190*x^36 + 223064037*x^35 + 751420521*x^34 - 4412144961*x^33 - 3725440704*x^32 + 54057041307*x^31 - 28786222110*x^30 - 426126343077*x^29 + 673008341841*x^28 + 2086637912508*x^27 - 5873721786378*x^26 - 5053254569676*x^25 + 30385493847345*x^24 - 5904503198856*x^23 - 98922911866245*x^22 + 93605350409913*x^21 + 191046717571311*x^20 - 349809055484526*x^19 - 147155959283342*x^18 + 707251464652203*x^17 - 217826629489476*x^16 - 797405893685064*x^15 + 705978096992721*x^14 + 378713501361567*x^13 - 782478874269585*x^12 + 139525392055671*x^11 + 385252617992454*x^10 - 250562776470715*x^9 - 36607492213488*x^8 + 95351499735039*x^7 - 31186566059442*x^6 - 5609742817023*x^5 + 6576945066567*x^4 - 1857401457582*x^3 + 224711418717*x^2 - 8430571116*x - 155181097)
 

\( x^{45} - 9 x^{44} - 117 x^{43} + 1356 x^{42} + 4455 x^{41} - 86382 x^{40} + 4596 x^{39} + 3024342 x^{38} - 5886234 x^{37} - 62776190 x^{36} + 223064037 x^{35} + \cdots - 155181097 \) 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:   \(148\!\cdots\!969\) \(\medspace = 3^{110}\cdot 31^{36}\) 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:  \(228.76\)
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:   \(3\), \(31\) 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:  \(837=3^{3}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{837}(256,·)$, $\chi_{837}(1,·)$, $\chi_{837}(4,·)$, $\chi_{837}(652,·)$, $\chi_{837}(655,·)$, $\chi_{837}(16,·)$, $\chi_{837}(529,·)$, $\chi_{837}(535,·)$, $\chi_{837}(280,·)$, $\chi_{837}(388,·)$, $\chi_{837}(283,·)$, $\chi_{837}(157,·)$, $\chi_{837}(376,·)$, $\chi_{837}(667,·)$, $\chi_{837}(295,·)$, $\chi_{837}(808,·)$, $\chi_{837}(814,·)$, $\chi_{837}(559,·)$, $\chi_{837}(562,·)$, $\chi_{837}(436,·)$, $\chi_{837}(442,·)$, $\chi_{837}(187,·)$, $\chi_{837}(574,·)$, $\chi_{837}(373,·)$, $\chi_{837}(64,·)$, $\chi_{837}(70,·)$, $\chi_{837}(97,·)$, $\chi_{837}(202,·)$, $\chi_{837}(715,·)$, $\chi_{837}(721,·)$, $\chi_{837}(466,·)$, $\chi_{837}(163,·)$, $\chi_{837}(469,·)$, $\chi_{837}(343,·)$, $\chi_{837}(349,·)$, $\chi_{837}(94,·)$, $\chi_{837}(481,·)$, $\chi_{837}(745,·)$, $\chi_{837}(748,·)$, $\chi_{837}(109,·)$, $\chi_{837}(622,·)$, $\chi_{837}(628,·)$, $\chi_{837}(190,·)$, $\chi_{837}(760,·)$, $\chi_{837}(250,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{5}a^{36}-\frac{2}{5}a^{35}-\frac{1}{5}a^{34}-\frac{2}{5}a^{33}+\frac{1}{5}a^{32}-\frac{2}{5}a^{31}-\frac{1}{5}a^{30}+\frac{2}{5}a^{29}-\frac{1}{5}a^{28}+\frac{2}{5}a^{27}+\frac{1}{5}a^{26}-\frac{1}{5}a^{25}-\frac{1}{5}a^{24}-\frac{2}{5}a^{23}+\frac{1}{5}a^{22}+\frac{2}{5}a^{21}+\frac{2}{5}a^{20}+\frac{2}{5}a^{19}+\frac{1}{5}a^{18}+\frac{1}{5}a^{17}-\frac{1}{5}a^{16}+\frac{2}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}-\frac{2}{5}a^{12}+\frac{2}{5}a^{11}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{37}+\frac{1}{5}a^{34}+\frac{2}{5}a^{33}-\frac{2}{5}a^{29}+\frac{1}{5}a^{26}+\frac{2}{5}a^{25}+\frac{1}{5}a^{24}+\frac{2}{5}a^{23}-\frac{1}{5}a^{22}+\frac{1}{5}a^{21}+\frac{1}{5}a^{20}-\frac{2}{5}a^{18}+\frac{1}{5}a^{17}-\frac{2}{5}a^{15}+\frac{1}{5}a^{14}-\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{38}+\frac{1}{5}a^{35}+\frac{2}{5}a^{34}-\frac{2}{5}a^{30}+\frac{1}{5}a^{27}+\frac{2}{5}a^{26}+\frac{1}{5}a^{25}+\frac{2}{5}a^{24}-\frac{1}{5}a^{23}+\frac{1}{5}a^{22}+\frac{1}{5}a^{21}-\frac{2}{5}a^{19}+\frac{1}{5}a^{18}-\frac{2}{5}a^{16}+\frac{1}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{5}a^{39}-\frac{1}{5}a^{35}+\frac{1}{5}a^{34}+\frac{2}{5}a^{33}-\frac{1}{5}a^{32}+\frac{1}{5}a^{30}-\frac{2}{5}a^{29}+\frac{2}{5}a^{28}-\frac{2}{5}a^{25}-\frac{2}{5}a^{23}-\frac{2}{5}a^{21}+\frac{1}{5}a^{20}-\frac{1}{5}a^{19}-\frac{1}{5}a^{18}+\frac{2}{5}a^{17}+\frac{2}{5}a^{16}+\frac{2}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{2}{5}a^{11}+\frac{2}{5}a^{10}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{40}-\frac{1}{5}a^{35}+\frac{1}{5}a^{34}+\frac{2}{5}a^{33}+\frac{1}{5}a^{32}-\frac{1}{5}a^{31}+\frac{2}{5}a^{30}-\frac{1}{5}a^{29}-\frac{1}{5}a^{28}+\frac{2}{5}a^{27}-\frac{1}{5}a^{26}-\frac{1}{5}a^{25}+\frac{2}{5}a^{24}-\frac{2}{5}a^{23}-\frac{1}{5}a^{22}-\frac{2}{5}a^{21}+\frac{1}{5}a^{20}+\frac{1}{5}a^{19}-\frac{2}{5}a^{18}-\frac{2}{5}a^{17}+\frac{1}{5}a^{16}+\frac{1}{5}a^{15}+\frac{2}{5}a^{14}-\frac{2}{5}a^{13}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{41}-\frac{1}{5}a^{35}+\frac{1}{5}a^{34}-\frac{1}{5}a^{33}-\frac{2}{5}a^{30}+\frac{1}{5}a^{29}+\frac{1}{5}a^{28}+\frac{1}{5}a^{27}+\frac{1}{5}a^{25}+\frac{2}{5}a^{24}+\frac{2}{5}a^{23}-\frac{1}{5}a^{22}-\frac{2}{5}a^{21}-\frac{2}{5}a^{20}-\frac{1}{5}a^{18}+\frac{2}{5}a^{17}-\frac{1}{5}a^{15}+\frac{2}{5}a^{14}-\frac{2}{5}a^{13}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{2536915}a^{42}-\frac{225354}{2536915}a^{41}-\frac{250386}{2536915}a^{40}+\frac{178183}{2536915}a^{39}-\frac{116036}{2536915}a^{38}-\frac{2993}{507383}a^{37}-\frac{79711}{2536915}a^{36}+\frac{467062}{2536915}a^{35}+\frac{125147}{507383}a^{34}-\frac{211527}{2536915}a^{33}+\frac{502301}{2536915}a^{32}-\frac{920006}{2536915}a^{31}-\frac{918368}{2536915}a^{30}-\frac{384288}{2536915}a^{29}+\frac{15494}{2536915}a^{28}+\frac{216943}{2536915}a^{27}+\frac{78881}{507383}a^{26}-\frac{866433}{2536915}a^{25}-\frac{56657}{507383}a^{24}-\frac{1218792}{2536915}a^{23}-\frac{870953}{2536915}a^{22}-\frac{1205484}{2536915}a^{21}-\frac{234942}{507383}a^{20}+\frac{702932}{2536915}a^{19}+\frac{690114}{2536915}a^{18}+\frac{77003}{507383}a^{17}+\frac{614031}{2536915}a^{16}-\frac{31239}{507383}a^{15}-\frac{1119864}{2536915}a^{14}+\frac{808968}{2536915}a^{13}+\frac{245050}{507383}a^{12}-\frac{197956}{2536915}a^{11}-\frac{518826}{2536915}a^{10}-\frac{1002586}{2536915}a^{9}+\frac{322329}{2536915}a^{8}+\frac{651611}{2536915}a^{7}+\frac{416507}{2536915}a^{6}+\frac{906634}{2536915}a^{5}-\frac{1153298}{2536915}a^{4}+\frac{475007}{2536915}a^{3}-\frac{222221}{507383}a^{2}+\frac{29253}{2536915}a+\frac{428487}{2536915}$, $\frac{1}{947991860285}a^{43}-\frac{143333}{947991860285}a^{42}+\frac{50939682921}{947991860285}a^{41}+\frac{45754886559}{947991860285}a^{40}-\frac{4761287697}{189598372057}a^{39}-\frac{66891765978}{947991860285}a^{38}-\frac{57478540271}{947991860285}a^{37}-\frac{9603924160}{189598372057}a^{36}-\frac{347515677589}{947991860285}a^{35}-\frac{268974497034}{947991860285}a^{34}-\frac{10976359285}{189598372057}a^{33}+\frac{122395319188}{947991860285}a^{32}+\frac{261951501421}{947991860285}a^{31}-\frac{247909636551}{947991860285}a^{30}-\frac{356413749093}{947991860285}a^{29}+\frac{359930960123}{947991860285}a^{28}-\frac{196378684071}{947991860285}a^{27}-\frac{91691532787}{189598372057}a^{26}+\frac{326833404499}{947991860285}a^{25}-\frac{102047831784}{947991860285}a^{24}+\frac{40883125981}{947991860285}a^{23}+\frac{236702650382}{947991860285}a^{22}-\frac{3545778507}{189598372057}a^{21}+\frac{366817937253}{947991860285}a^{20}+\frac{351675191292}{947991860285}a^{19}+\frac{330739708628}{947991860285}a^{18}+\frac{191702900784}{947991860285}a^{17}+\frac{334406938899}{947991860285}a^{16}-\frac{433120349031}{947991860285}a^{15}-\frac{76805408392}{189598372057}a^{14}+\frac{212291147102}{947991860285}a^{13}-\frac{168409854344}{947991860285}a^{12}+\frac{185528855878}{947991860285}a^{11}+\frac{279301253969}{947991860285}a^{10}-\frac{115812405683}{947991860285}a^{9}-\frac{16574487922}{947991860285}a^{8}-\frac{261231212113}{947991860285}a^{7}+\frac{450286291369}{947991860285}a^{6}+\frac{335981740774}{947991860285}a^{5}-\frac{441713663722}{947991860285}a^{4}+\frac{273966559067}{947991860285}a^{3}+\frac{82019160309}{947991860285}a^{2}+\frac{4258332729}{947991860285}a-\frac{382130398654}{947991860285}$, $\frac{1}{21\!\cdots\!45}a^{44}-\frac{61\!\cdots\!09}{43\!\cdots\!69}a^{43}-\frac{41\!\cdots\!98}{21\!\cdots\!45}a^{42}-\frac{74\!\cdots\!88}{21\!\cdots\!45}a^{41}-\frac{53\!\cdots\!64}{21\!\cdots\!45}a^{40}-\frac{33\!\cdots\!61}{43\!\cdots\!69}a^{39}+\frac{77\!\cdots\!63}{21\!\cdots\!45}a^{38}+\frac{11\!\cdots\!62}{21\!\cdots\!45}a^{37}+\frac{81\!\cdots\!01}{21\!\cdots\!45}a^{36}-\frac{54\!\cdots\!00}{43\!\cdots\!69}a^{35}-\frac{98\!\cdots\!24}{21\!\cdots\!45}a^{34}-\frac{94\!\cdots\!48}{21\!\cdots\!45}a^{33}+\frac{10\!\cdots\!76}{21\!\cdots\!45}a^{32}+\frac{69\!\cdots\!28}{21\!\cdots\!45}a^{31}+\frac{51\!\cdots\!99}{21\!\cdots\!45}a^{30}+\frac{35\!\cdots\!98}{50\!\cdots\!65}a^{29}-\frac{12\!\cdots\!36}{43\!\cdots\!69}a^{28}+\frac{21\!\cdots\!68}{21\!\cdots\!45}a^{27}-\frac{47\!\cdots\!44}{21\!\cdots\!45}a^{26}+\frac{61\!\cdots\!91}{21\!\cdots\!45}a^{25}-\frac{87\!\cdots\!52}{21\!\cdots\!45}a^{24}-\frac{44\!\cdots\!27}{21\!\cdots\!45}a^{23}-\frac{21\!\cdots\!42}{43\!\cdots\!69}a^{22}-\frac{48\!\cdots\!78}{21\!\cdots\!45}a^{21}-\frac{49\!\cdots\!12}{21\!\cdots\!45}a^{20}-\frac{77\!\cdots\!46}{21\!\cdots\!45}a^{19}+\frac{73\!\cdots\!40}{43\!\cdots\!69}a^{18}+\frac{35\!\cdots\!74}{21\!\cdots\!45}a^{17}+\frac{12\!\cdots\!59}{43\!\cdots\!69}a^{16}-\frac{38\!\cdots\!68}{43\!\cdots\!69}a^{15}-\frac{10\!\cdots\!16}{21\!\cdots\!45}a^{14}-\frac{55\!\cdots\!78}{21\!\cdots\!45}a^{13}-\frac{39\!\cdots\!21}{21\!\cdots\!45}a^{12}+\frac{61\!\cdots\!39}{21\!\cdots\!45}a^{11}+\frac{21\!\cdots\!26}{43\!\cdots\!69}a^{10}+\frac{56\!\cdots\!02}{21\!\cdots\!45}a^{9}-\frac{67\!\cdots\!31}{21\!\cdots\!45}a^{8}+\frac{80\!\cdots\!29}{21\!\cdots\!45}a^{7}+\frac{48\!\cdots\!93}{21\!\cdots\!45}a^{6}+\frac{72\!\cdots\!03}{21\!\cdots\!45}a^{5}-\frac{36\!\cdots\!58}{21\!\cdots\!45}a^{4}-\frac{48\!\cdots\!56}{21\!\cdots\!45}a^{3}-\frac{56\!\cdots\!41}{21\!\cdots\!45}a^{2}-\frac{63\!\cdots\!94}{21\!\cdots\!45}a+\frac{66\!\cdots\!14}{21\!\cdots\!45}$ 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 - 9*x^44 - 117*x^43 + 1356*x^42 + 4455*x^41 - 86382*x^40 + 4596*x^39 + 3024342*x^38 - 5886234*x^37 - 62776190*x^36 + 223064037*x^35 + 751420521*x^34 - 4412144961*x^33 - 3725440704*x^32 + 54057041307*x^31 - 28786222110*x^30 - 426126343077*x^29 + 673008341841*x^28 + 2086637912508*x^27 - 5873721786378*x^26 - 5053254569676*x^25 + 30385493847345*x^24 - 5904503198856*x^23 - 98922911866245*x^22 + 93605350409913*x^21 + 191046717571311*x^20 - 349809055484526*x^19 - 147155959283342*x^18 + 707251464652203*x^17 - 217826629489476*x^16 - 797405893685064*x^15 + 705978096992721*x^14 + 378713501361567*x^13 - 782478874269585*x^12 + 139525392055671*x^11 + 385252617992454*x^10 - 250562776470715*x^9 - 36607492213488*x^8 + 95351499735039*x^7 - 31186566059442*x^6 - 5609742817023*x^5 + 6576945066567*x^4 - 1857401457582*x^3 + 224711418717*x^2 - 8430571116*x - 155181097)
 
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 - 9*x^44 - 117*x^43 + 1356*x^42 + 4455*x^41 - 86382*x^40 + 4596*x^39 + 3024342*x^38 - 5886234*x^37 - 62776190*x^36 + 223064037*x^35 + 751420521*x^34 - 4412144961*x^33 - 3725440704*x^32 + 54057041307*x^31 - 28786222110*x^30 - 426126343077*x^29 + 673008341841*x^28 + 2086637912508*x^27 - 5873721786378*x^26 - 5053254569676*x^25 + 30385493847345*x^24 - 5904503198856*x^23 - 98922911866245*x^22 + 93605350409913*x^21 + 191046717571311*x^20 - 349809055484526*x^19 - 147155959283342*x^18 + 707251464652203*x^17 - 217826629489476*x^16 - 797405893685064*x^15 + 705978096992721*x^14 + 378713501361567*x^13 - 782478874269585*x^12 + 139525392055671*x^11 + 385252617992454*x^10 - 250562776470715*x^9 - 36607492213488*x^8 + 95351499735039*x^7 - 31186566059442*x^6 - 5609742817023*x^5 + 6576945066567*x^4 - 1857401457582*x^3 + 224711418717*x^2 - 8430571116*x - 155181097, 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 - 9*x^44 - 117*x^43 + 1356*x^42 + 4455*x^41 - 86382*x^40 + 4596*x^39 + 3024342*x^38 - 5886234*x^37 - 62776190*x^36 + 223064037*x^35 + 751420521*x^34 - 4412144961*x^33 - 3725440704*x^32 + 54057041307*x^31 - 28786222110*x^30 - 426126343077*x^29 + 673008341841*x^28 + 2086637912508*x^27 - 5873721786378*x^26 - 5053254569676*x^25 + 30385493847345*x^24 - 5904503198856*x^23 - 98922911866245*x^22 + 93605350409913*x^21 + 191046717571311*x^20 - 349809055484526*x^19 - 147155959283342*x^18 + 707251464652203*x^17 - 217826629489476*x^16 - 797405893685064*x^15 + 705978096992721*x^14 + 378713501361567*x^13 - 782478874269585*x^12 + 139525392055671*x^11 + 385252617992454*x^10 - 250562776470715*x^9 - 36607492213488*x^8 + 95351499735039*x^7 - 31186566059442*x^6 - 5609742817023*x^5 + 6576945066567*x^4 - 1857401457582*x^3 + 224711418717*x^2 - 8430571116*x - 155181097);
 
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 - 9*x^44 - 117*x^43 + 1356*x^42 + 4455*x^41 - 86382*x^40 + 4596*x^39 + 3024342*x^38 - 5886234*x^37 - 62776190*x^36 + 223064037*x^35 + 751420521*x^34 - 4412144961*x^33 - 3725440704*x^32 + 54057041307*x^31 - 28786222110*x^30 - 426126343077*x^29 + 673008341841*x^28 + 2086637912508*x^27 - 5873721786378*x^26 - 5053254569676*x^25 + 30385493847345*x^24 - 5904503198856*x^23 - 98922911866245*x^22 + 93605350409913*x^21 + 191046717571311*x^20 - 349809055484526*x^19 - 147155959283342*x^18 + 707251464652203*x^17 - 217826629489476*x^16 - 797405893685064*x^15 + 705978096992721*x^14 + 378713501361567*x^13 - 782478874269585*x^12 + 139525392055671*x^11 + 385252617992454*x^10 - 250562776470715*x^9 - 36607492213488*x^8 + 95351499735039*x^7 - 31186566059442*x^6 - 5609742817023*x^5 + 6576945066567*x^4 - 1857401457582*x^3 + 224711418717*x^2 - 8430571116*x - 155181097);
 
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

\(\Q(\zeta_{9})^+\), 5.5.923521.1, \(\Q(\zeta_{27})^+\), 15.15.2746410307762150989067078161.1

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 $45$ R ${\href{/padicField/5.9.0.1}{9} }^{5}$ $45$ $45$ $45$ $15^{3}$ $15^{3}$ $45$ $45$ R ${\href{/padicField/37.3.0.1}{3} }^{15}$ $45$ $45$ $45$ ${\href{/padicField/53.5.0.1}{5} }^{9}$ $45$

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
\(3\) Copy content Toggle raw display Deg $45$$9$$5$$110$
\(31\) Copy content Toggle raw display Deg $45$$5$$9$$36$