Properties

Label 45.45.612...361.1
Degree $45$
Signature $[45, 0]$
Discriminant $6.128\times 10^{113}$
Root discriminant \(337.76\)
Ramified primes $19,31$
Class number not computed
Class group not computed
Galois group $C_{45}$ (as 45T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^45 - x^44 - 250*x^43 - 57*x^42 + 27766*x^41 + 34148*x^40 - 1801054*x^39 - 3701713*x^38 + 75995397*x^37 + 210980315*x^36 - 2204269403*x^35 - 7656169786*x^34 + 45208500184*x^33 + 192019854445*x^32 - 660457840142*x^31 - 3478319662546*x^30 + 6740250648134*x^29 + 46682393472693*x^28 - 44214268231149*x^27 - 471014331370572*x^26 + 119457837724165*x^25 + 3599175944415558*x^24 + 920658436866528*x^23 - 20872201046519003*x^22 - 13494994939768731*x^21 + 91678287510298882*x^20 + 87997267245853232*x^19 - 303360026160716188*x^18 - 371303902979085816*x^17 + 749732931610001378*x^16 + 1094855896004641434*x^15 - 1366727950240442050*x^14 - 2306120975093560007*x^13 + 1803267431605382487*x^12 + 3471556977526533885*x^11 - 1666809346078423573*x^10 - 3682047656706158942*x^9 + 1008367868417943528*x^8 + 2668021411897921137*x^7 - 327662726329807390*x^6 - 1248245094003212175*x^5 + 152924933759221*x^4 + 338163088283119511*x^3 + 38015736547359269*x^2 - 40401517332214916*x - 8937046621536643)
 
Copy content gp:K = bnfinit(y^45 - y^44 - 250*y^43 - 57*y^42 + 27766*y^41 + 34148*y^40 - 1801054*y^39 - 3701713*y^38 + 75995397*y^37 + 210980315*y^36 - 2204269403*y^35 - 7656169786*y^34 + 45208500184*y^33 + 192019854445*y^32 - 660457840142*y^31 - 3478319662546*y^30 + 6740250648134*y^29 + 46682393472693*y^28 - 44214268231149*y^27 - 471014331370572*y^26 + 119457837724165*y^25 + 3599175944415558*y^24 + 920658436866528*y^23 - 20872201046519003*y^22 - 13494994939768731*y^21 + 91678287510298882*y^20 + 87997267245853232*y^19 - 303360026160716188*y^18 - 371303902979085816*y^17 + 749732931610001378*y^16 + 1094855896004641434*y^15 - 1366727950240442050*y^14 - 2306120975093560007*y^13 + 1803267431605382487*y^12 + 3471556977526533885*y^11 - 1666809346078423573*y^10 - 3682047656706158942*y^9 + 1008367868417943528*y^8 + 2668021411897921137*y^7 - 327662726329807390*y^6 - 1248245094003212175*y^5 + 152924933759221*y^4 + 338163088283119511*y^3 + 38015736547359269*y^2 - 40401517332214916*y - 8937046621536643, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - x^44 - 250*x^43 - 57*x^42 + 27766*x^41 + 34148*x^40 - 1801054*x^39 - 3701713*x^38 + 75995397*x^37 + 210980315*x^36 - 2204269403*x^35 - 7656169786*x^34 + 45208500184*x^33 + 192019854445*x^32 - 660457840142*x^31 - 3478319662546*x^30 + 6740250648134*x^29 + 46682393472693*x^28 - 44214268231149*x^27 - 471014331370572*x^26 + 119457837724165*x^25 + 3599175944415558*x^24 + 920658436866528*x^23 - 20872201046519003*x^22 - 13494994939768731*x^21 + 91678287510298882*x^20 + 87997267245853232*x^19 - 303360026160716188*x^18 - 371303902979085816*x^17 + 749732931610001378*x^16 + 1094855896004641434*x^15 - 1366727950240442050*x^14 - 2306120975093560007*x^13 + 1803267431605382487*x^12 + 3471556977526533885*x^11 - 1666809346078423573*x^10 - 3682047656706158942*x^9 + 1008367868417943528*x^8 + 2668021411897921137*x^7 - 327662726329807390*x^6 - 1248245094003212175*x^5 + 152924933759221*x^4 + 338163088283119511*x^3 + 38015736547359269*x^2 - 40401517332214916*x - 8937046621536643);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^45 - x^44 - 250*x^43 - 57*x^42 + 27766*x^41 + 34148*x^40 - 1801054*x^39 - 3701713*x^38 + 75995397*x^37 + 210980315*x^36 - 2204269403*x^35 - 7656169786*x^34 + 45208500184*x^33 + 192019854445*x^32 - 660457840142*x^31 - 3478319662546*x^30 + 6740250648134*x^29 + 46682393472693*x^28 - 44214268231149*x^27 - 471014331370572*x^26 + 119457837724165*x^25 + 3599175944415558*x^24 + 920658436866528*x^23 - 20872201046519003*x^22 - 13494994939768731*x^21 + 91678287510298882*x^20 + 87997267245853232*x^19 - 303360026160716188*x^18 - 371303902979085816*x^17 + 749732931610001378*x^16 + 1094855896004641434*x^15 - 1366727950240442050*x^14 - 2306120975093560007*x^13 + 1803267431605382487*x^12 + 3471556977526533885*x^11 - 1666809346078423573*x^10 - 3682047656706158942*x^9 + 1008367868417943528*x^8 + 2668021411897921137*x^7 - 327662726329807390*x^6 - 1248245094003212175*x^5 + 152924933759221*x^4 + 338163088283119511*x^3 + 38015736547359269*x^2 - 40401517332214916*x - 8937046621536643)
 

\( x^{45} - x^{44} - 250 x^{43} - 57 x^{42} + 27766 x^{41} + 34148 x^{40} - 1801054 x^{39} + \cdots - 89\!\cdots\!43 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $45$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[45, 0]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(612\!\cdots\!361\) \(\medspace = 19^{40}\cdot 31^{42}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(337.76\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $19^{8/9}31^{14/15}\approx 337.75940307002355$
Ramified primes:   \(19\), \(31\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{45}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(589=19\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{589}(1,·)$, $\chi_{589}(131,·)$, $\chi_{589}(9,·)$, $\chi_{589}(138,·)$, $\chi_{589}(140,·)$, $\chi_{589}(397,·)$, $\chi_{589}(142,·)$, $\chi_{589}(536,·)$, $\chi_{589}(149,·)$, $\chi_{589}(408,·)$, $\chi_{589}(543,·)$, $\chi_{589}(289,·)$, $\chi_{589}(419,·)$, $\chi_{589}(39,·)$, $\chi_{589}(169,·)$, $\chi_{589}(175,·)$, $\chi_{589}(562,·)$, $\chi_{589}(438,·)$, $\chi_{589}(311,·)$, $\chi_{589}(159,·)$, $\chi_{589}(64,·)$, $\chi_{589}(196,·)$, $\chi_{589}(453,·)$, $\chi_{589}(586,·)$, $\chi_{589}(80,·)$, $\chi_{589}(81,·)$, $\chi_{589}(82,·)$, $\chi_{589}(163,·)$, $\chi_{589}(214,·)$, $\chi_{589}(343,·)$, $\chi_{589}(472,·)$, $\chi_{589}(346,·)$, $\chi_{589}(349,·)$, $\chi_{589}(351,·)$, $\chi_{589}(443,·)$, $\chi_{589}(100,·)$, $\chi_{589}(237,·)$, $\chi_{589}(366,·)$, $\chi_{589}(125,·)$, $\chi_{589}(112,·)$, $\chi_{589}(467,·)$, $\chi_{589}(245,·)$, $\chi_{589}(576,·)$, $\chi_{589}(253,·)$, $\chi_{589}(510,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{563}a^{42}+\frac{232}{563}a^{41}-\frac{61}{563}a^{40}-\frac{120}{563}a^{39}-\frac{20}{563}a^{38}+\frac{240}{563}a^{37}+\frac{122}{563}a^{36}-\frac{153}{563}a^{35}+\frac{80}{563}a^{34}-\frac{184}{563}a^{33}-\frac{54}{563}a^{32}-\frac{198}{563}a^{31}-\frac{66}{563}a^{30}+\frac{177}{563}a^{29}-\frac{47}{563}a^{28}-\frac{200}{563}a^{27}-\frac{28}{563}a^{26}+\frac{151}{563}a^{25}-\frac{50}{563}a^{24}+\frac{122}{563}a^{23}+\frac{87}{563}a^{22}+\frac{170}{563}a^{21}-\frac{71}{563}a^{20}-\frac{245}{563}a^{19}-\frac{34}{563}a^{18}+\frac{142}{563}a^{17}-\frac{37}{563}a^{16}-\frac{142}{563}a^{15}-\frac{82}{563}a^{14}+\frac{82}{563}a^{13}-\frac{225}{563}a^{12}+\frac{235}{563}a^{11}+\frac{262}{563}a^{10}+\frac{67}{563}a^{9}-\frac{146}{563}a^{8}-\frac{5}{563}a^{7}+\frac{237}{563}a^{6}-\frac{28}{563}a^{5}+\frac{182}{563}a^{4}+\frac{155}{563}a^{3}+\frac{163}{563}a^{2}-\frac{44}{563}a-\frac{196}{563}$, $\frac{1}{563}a^{43}+\frac{163}{563}a^{41}-\frac{43}{563}a^{40}+\frac{233}{563}a^{39}-\frac{187}{563}a^{38}+\frac{179}{563}a^{37}+\frac{256}{563}a^{36}+\frac{107}{563}a^{35}-\frac{165}{563}a^{34}-\frac{154}{563}a^{33}-\frac{56}{563}a^{32}+\frac{267}{563}a^{31}-\frac{275}{563}a^{30}-\frac{12}{563}a^{29}+\frac{7}{563}a^{28}+\frac{206}{563}a^{27}-\frac{109}{563}a^{26}-\frac{176}{563}a^{25}-\frac{101}{563}a^{24}-\frac{67}{563}a^{23}+\frac{254}{563}a^{22}-\frac{101}{563}a^{21}-\frac{100}{563}a^{20}-\frac{57}{563}a^{19}+\frac{148}{563}a^{18}+\frac{236}{563}a^{17}-\frac{3}{563}a^{16}+\frac{208}{563}a^{15}-\frac{36}{563}a^{14}-\frac{107}{563}a^{13}+\frac{76}{563}a^{12}-\frac{210}{563}a^{11}+\frac{87}{563}a^{10}+\frac{74}{563}a^{9}+\frac{87}{563}a^{8}+\frac{271}{563}a^{7}+\frac{162}{563}a^{6}-\frac{78}{563}a^{5}+\frac{156}{563}a^{4}+\frac{235}{563}a^{3}-\frac{139}{563}a^{2}-\frac{122}{563}a-\frac{131}{563}$, $\frac{1}{59\cdots 99}a^{44}-\frac{48\cdots 39}{59\cdots 99}a^{43}+\frac{50\cdots 37}{59\cdots 99}a^{42}+\frac{15\cdots 07}{59\cdots 99}a^{41}+\frac{35\cdots 30}{59\cdots 99}a^{40}+\frac{21\cdots 38}{59\cdots 99}a^{39}+\frac{18\cdots 94}{59\cdots 99}a^{38}+\frac{25\cdots 13}{59\cdots 99}a^{37}-\frac{25\cdots 34}{59\cdots 99}a^{36}-\frac{18\cdots 41}{59\cdots 99}a^{35}+\frac{26\cdots 18}{59\cdots 99}a^{34}+\frac{26\cdots 64}{59\cdots 99}a^{33}-\frac{29\cdots 44}{59\cdots 99}a^{32}+\frac{81\cdots 68}{59\cdots 99}a^{31}-\frac{95\cdots 62}{59\cdots 99}a^{30}-\frac{13\cdots 81}{59\cdots 99}a^{29}+\frac{23\cdots 19}{59\cdots 99}a^{28}+\frac{15\cdots 17}{59\cdots 99}a^{27}-\frac{21\cdots 04}{59\cdots 99}a^{26}+\frac{19\cdots 68}{59\cdots 99}a^{25}-\frac{53\cdots 28}{59\cdots 99}a^{24}+\frac{17\cdots 35}{59\cdots 99}a^{23}+\frac{11\cdots 45}{59\cdots 99}a^{22}+\frac{21\cdots 24}{59\cdots 99}a^{21}-\frac{14\cdots 53}{59\cdots 99}a^{20}-\frac{15\cdots 78}{59\cdots 99}a^{19}+\frac{18\cdots 97}{59\cdots 99}a^{18}+\frac{28\cdots 84}{59\cdots 99}a^{17}+\frac{82\cdots 99}{59\cdots 99}a^{16}-\frac{29\cdots 01}{59\cdots 99}a^{15}-\frac{27\cdots 47}{59\cdots 99}a^{14}-\frac{74\cdots 48}{59\cdots 99}a^{13}+\frac{28\cdots 25}{59\cdots 99}a^{12}+\frac{13\cdots 88}{59\cdots 99}a^{11}+\frac{10\cdots 16}{59\cdots 99}a^{10}+\frac{21\cdots 81}{59\cdots 99}a^{9}+\frac{14\cdots 33}{59\cdots 99}a^{8}-\frac{80\cdots 06}{59\cdots 99}a^{7}+\frac{24\cdots 99}{59\cdots 99}a^{6}+\frac{39\cdots 38}{59\cdots 99}a^{5}+\frac{14\cdots 62}{59\cdots 99}a^{4}-\frac{19\cdots 08}{59\cdots 99}a^{3}+\frac{13\cdots 35}{59\cdots 99}a^{2}-\frac{23\cdots 47}{59\cdots 99}a+\frac{94\cdots 39}{59\cdots 99}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  not computed
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $44$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:  not computed
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{45}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{612823173021751971297676000847719042376493061735873445231828455051262946540529457157884846548736505117837633099361}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^45 - x^44 - 250*x^43 - 57*x^42 + 27766*x^41 + 34148*x^40 - 1801054*x^39 - 3701713*x^38 + 75995397*x^37 + 210980315*x^36 - 2204269403*x^35 - 7656169786*x^34 + 45208500184*x^33 + 192019854445*x^32 - 660457840142*x^31 - 3478319662546*x^30 + 6740250648134*x^29 + 46682393472693*x^28 - 44214268231149*x^27 - 471014331370572*x^26 + 119457837724165*x^25 + 3599175944415558*x^24 + 920658436866528*x^23 - 20872201046519003*x^22 - 13494994939768731*x^21 + 91678287510298882*x^20 + 87997267245853232*x^19 - 303360026160716188*x^18 - 371303902979085816*x^17 + 749732931610001378*x^16 + 1094855896004641434*x^15 - 1366727950240442050*x^14 - 2306120975093560007*x^13 + 1803267431605382487*x^12 + 3471556977526533885*x^11 - 1666809346078423573*x^10 - 3682047656706158942*x^9 + 1008367868417943528*x^8 + 2668021411897921137*x^7 - 327662726329807390*x^6 - 1248245094003212175*x^5 + 152924933759221*x^4 + 338163088283119511*x^3 + 38015736547359269*x^2 - 40401517332214916*x - 8937046621536643) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^45 - x^44 - 250*x^43 - 57*x^42 + 27766*x^41 + 34148*x^40 - 1801054*x^39 - 3701713*x^38 + 75995397*x^37 + 210980315*x^36 - 2204269403*x^35 - 7656169786*x^34 + 45208500184*x^33 + 192019854445*x^32 - 660457840142*x^31 - 3478319662546*x^30 + 6740250648134*x^29 + 46682393472693*x^28 - 44214268231149*x^27 - 471014331370572*x^26 + 119457837724165*x^25 + 3599175944415558*x^24 + 920658436866528*x^23 - 20872201046519003*x^22 - 13494994939768731*x^21 + 91678287510298882*x^20 + 87997267245853232*x^19 - 303360026160716188*x^18 - 371303902979085816*x^17 + 749732931610001378*x^16 + 1094855896004641434*x^15 - 1366727950240442050*x^14 - 2306120975093560007*x^13 + 1803267431605382487*x^12 + 3471556977526533885*x^11 - 1666809346078423573*x^10 - 3682047656706158942*x^9 + 1008367868417943528*x^8 + 2668021411897921137*x^7 - 327662726329807390*x^6 - 1248245094003212175*x^5 + 152924933759221*x^4 + 338163088283119511*x^3 + 38015736547359269*x^2 - 40401517332214916*x - 8937046621536643, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - x^44 - 250*x^43 - 57*x^42 + 27766*x^41 + 34148*x^40 - 1801054*x^39 - 3701713*x^38 + 75995397*x^37 + 210980315*x^36 - 2204269403*x^35 - 7656169786*x^34 + 45208500184*x^33 + 192019854445*x^32 - 660457840142*x^31 - 3478319662546*x^30 + 6740250648134*x^29 + 46682393472693*x^28 - 44214268231149*x^27 - 471014331370572*x^26 + 119457837724165*x^25 + 3599175944415558*x^24 + 920658436866528*x^23 - 20872201046519003*x^22 - 13494994939768731*x^21 + 91678287510298882*x^20 + 87997267245853232*x^19 - 303360026160716188*x^18 - 371303902979085816*x^17 + 749732931610001378*x^16 + 1094855896004641434*x^15 - 1366727950240442050*x^14 - 2306120975093560007*x^13 + 1803267431605382487*x^12 + 3471556977526533885*x^11 - 1666809346078423573*x^10 - 3682047656706158942*x^9 + 1008367868417943528*x^8 + 2668021411897921137*x^7 - 327662726329807390*x^6 - 1248245094003212175*x^5 + 152924933759221*x^4 + 338163088283119511*x^3 + 38015736547359269*x^2 - 40401517332214916*x - 8937046621536643); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^45 - x^44 - 250*x^43 - 57*x^42 + 27766*x^41 + 34148*x^40 - 1801054*x^39 - 3701713*x^38 + 75995397*x^37 + 210980315*x^36 - 2204269403*x^35 - 7656169786*x^34 + 45208500184*x^33 + 192019854445*x^32 - 660457840142*x^31 - 3478319662546*x^30 + 6740250648134*x^29 + 46682393472693*x^28 - 44214268231149*x^27 - 471014331370572*x^26 + 119457837724165*x^25 + 3599175944415558*x^24 + 920658436866528*x^23 - 20872201046519003*x^22 - 13494994939768731*x^21 + 91678287510298882*x^20 + 87997267245853232*x^19 - 303360026160716188*x^18 - 371303902979085816*x^17 + 749732931610001378*x^16 + 1094855896004641434*x^15 - 1366727950240442050*x^14 - 2306120975093560007*x^13 + 1803267431605382487*x^12 + 3471556977526533885*x^11 - 1666809346078423573*x^10 - 3682047656706158942*x^9 + 1008367868417943528*x^8 + 2668021411897921137*x^7 - 327662726329807390*x^6 - 1248245094003212175*x^5 + 152924933759221*x^4 + 338163088283119511*x^3 + 38015736547359269*x^2 - 40401517332214916*x - 8937046621536643); 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):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A cyclic group of order 45
The 45 conjugacy class representatives for $C_{45}$
Character table for $C_{45}$

Intermediate fields

3.3.361.1, 5.5.923521.1, 9.9.15072974715383053921.2, 15.15.4829212716211581952447142935561.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content 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 $45$ $45$ ${\href{/padicField/5.9.0.1}{9} }^{5}$ $15^{3}$ $15^{3}$ $45$ $45$ R $45$ $45$ R ${\href{/padicField/37.3.0.1}{3} }^{15}$ $45$ $45$ $45$ $45$ $45$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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$
\(31\) Copy content Toggle raw display 31.1.15.14a1.12$x^{15} + 775$$15$$1$$14$$C_{15}$$$[\ ]_{15}$$
31.1.15.14a1.12$x^{15} + 775$$15$$1$$14$$C_{15}$$$[\ ]_{15}$$
31.1.15.14a1.12$x^{15} + 775$$15$$1$$14$$C_{15}$$$[\ ]_{15}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)