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Properties

Label 45.45.113...889.1
Degree $45$
Signature $[45, 0]$
Discriminant $1.136\times 10^{96}$
Root discriminant \(136.32\)
Ramified primes $13,31$
Class number not computed
Class group not computed
Galois group $C_3\times C_{15}$ (as 45T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 12*x^44 - 38*x^43 + 936*x^42 - 488*x^41 - 32994*x^40 + 61155*x^39 + 695151*x^38 - 1826452*x^37 - 9745559*x^36 + 31398650*x^35 + 95576968*x^34 - 363031278*x^33 - 668271620*x^32 + 3006554151*x^31 + 3299628259*x^30 - 18405469866*x^29 - 10819187452*x^28 + 84606290574*x^27 + 17798672756*x^26 - 293624042239*x^25 + 25779977473*x^24 + 766879229695*x^23 - 259704306954*x^22 - 1488735953755*x^21 + 843265985246*x^20 + 2094958274346*x^19 - 1672596038928*x^18 - 2036992655261*x^17 + 2211683013511*x^16 + 1229593352332*x^15 - 1960884055901*x^14 - 305227614428*x^13 + 1128154934100*x^12 - 126915280282*x^11 - 391023252143*x^10 + 124192084723*x^9 + 69369604379*x^8 - 37452923715*x^7 - 3334757683*x^6 + 4610823573*x^5 - 413774775*x^4 - 188088413*x^3 + 35065204*x^2 - 872335*x + 619)
 
gp: K = bnfinit(y^45 - 12*y^44 - 38*y^43 + 936*y^42 - 488*y^41 - 32994*y^40 + 61155*y^39 + 695151*y^38 - 1826452*y^37 - 9745559*y^36 + 31398650*y^35 + 95576968*y^34 - 363031278*y^33 - 668271620*y^32 + 3006554151*y^31 + 3299628259*y^30 - 18405469866*y^29 - 10819187452*y^28 + 84606290574*y^27 + 17798672756*y^26 - 293624042239*y^25 + 25779977473*y^24 + 766879229695*y^23 - 259704306954*y^22 - 1488735953755*y^21 + 843265985246*y^20 + 2094958274346*y^19 - 1672596038928*y^18 - 2036992655261*y^17 + 2211683013511*y^16 + 1229593352332*y^15 - 1960884055901*y^14 - 305227614428*y^13 + 1128154934100*y^12 - 126915280282*y^11 - 391023252143*y^10 + 124192084723*y^9 + 69369604379*y^8 - 37452923715*y^7 - 3334757683*y^6 + 4610823573*y^5 - 413774775*y^4 - 188088413*y^3 + 35065204*y^2 - 872335*y + 619, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 12*x^44 - 38*x^43 + 936*x^42 - 488*x^41 - 32994*x^40 + 61155*x^39 + 695151*x^38 - 1826452*x^37 - 9745559*x^36 + 31398650*x^35 + 95576968*x^34 - 363031278*x^33 - 668271620*x^32 + 3006554151*x^31 + 3299628259*x^30 - 18405469866*x^29 - 10819187452*x^28 + 84606290574*x^27 + 17798672756*x^26 - 293624042239*x^25 + 25779977473*x^24 + 766879229695*x^23 - 259704306954*x^22 - 1488735953755*x^21 + 843265985246*x^20 + 2094958274346*x^19 - 1672596038928*x^18 - 2036992655261*x^17 + 2211683013511*x^16 + 1229593352332*x^15 - 1960884055901*x^14 - 305227614428*x^13 + 1128154934100*x^12 - 126915280282*x^11 - 391023252143*x^10 + 124192084723*x^9 + 69369604379*x^8 - 37452923715*x^7 - 3334757683*x^6 + 4610823573*x^5 - 413774775*x^4 - 188088413*x^3 + 35065204*x^2 - 872335*x + 619);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 12*x^44 - 38*x^43 + 936*x^42 - 488*x^41 - 32994*x^40 + 61155*x^39 + 695151*x^38 - 1826452*x^37 - 9745559*x^36 + 31398650*x^35 + 95576968*x^34 - 363031278*x^33 - 668271620*x^32 + 3006554151*x^31 + 3299628259*x^30 - 18405469866*x^29 - 10819187452*x^28 + 84606290574*x^27 + 17798672756*x^26 - 293624042239*x^25 + 25779977473*x^24 + 766879229695*x^23 - 259704306954*x^22 - 1488735953755*x^21 + 843265985246*x^20 + 2094958274346*x^19 - 1672596038928*x^18 - 2036992655261*x^17 + 2211683013511*x^16 + 1229593352332*x^15 - 1960884055901*x^14 - 305227614428*x^13 + 1128154934100*x^12 - 126915280282*x^11 - 391023252143*x^10 + 124192084723*x^9 + 69369604379*x^8 - 37452923715*x^7 - 3334757683*x^6 + 4610823573*x^5 - 413774775*x^4 - 188088413*x^3 + 35065204*x^2 - 872335*x + 619)
 

\( x^{45} - 12 x^{44} - 38 x^{43} + 936 x^{42} - 488 x^{41} - 32994 x^{40} + 61155 x^{39} + 695151 x^{38} + \cdots + 619 \) 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:   \(113\!\cdots\!889\) \(\medspace = 13^{30}\cdot 31^{42}\) 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:  \(136.32\)
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))
 
Galois root discriminant:  $13^{2/3}31^{14/15}\approx 136.32216742518625$
Ramified primes:   \(13\), \(31\) 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:  \(403=13\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{403}(256,·)$, $\chi_{403}(1,·)$, $\chi_{403}(386,·)$, $\chi_{403}(131,·)$, $\chi_{403}(133,·)$, $\chi_{403}(391,·)$, $\chi_{403}(9,·)$, $\chi_{403}(14,·)$, $\chi_{403}(16,·)$, $\chi_{403}(107,·)$, $\chi_{403}(276,·)$, $\chi_{403}(152,·)$, $\chi_{403}(157,·)$, $\chi_{403}(287,·)$, $\chi_{403}(289,·)$, $\chi_{403}(35,·)$, $\chi_{403}(165,·)$, $\chi_{403}(295,·)$, $\chi_{403}(40,·)$, $\chi_{403}(94,·)$, $\chi_{403}(183,·)$, $\chi_{403}(159,·)$, $\chi_{403}(191,·)$, $\chi_{403}(66,·)$, $\chi_{403}(196,·)$, $\chi_{403}(326,·)$, $\chi_{403}(328,·)$, $\chi_{403}(204,·)$, $\chi_{403}(81,·)$, $\chi_{403}(211,·)$, $\chi_{403}(87,·)$, $\chi_{403}(222,·)$, $\chi_{403}(224,·)$, $\chi_{403}(144,·)$, $\chi_{403}(315,·)$, $\chi_{403}(100,·)$, $\chi_{403}(360,·)$, $\chi_{403}(235,·)$, $\chi_{403}(237,·)$, $\chi_{403}(113,·)$, $\chi_{403}(373,·)$, $\chi_{403}(118,·)$, $\chi_{403}(250,·)$, $\chi_{403}(380,·)$, $\chi_{403}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $\frac{1}{5}a^{39}+\frac{2}{5}a^{38}-\frac{1}{5}a^{36}-\frac{1}{5}a^{35}+\frac{1}{5}a^{34}+\frac{2}{5}a^{31}-\frac{1}{5}a^{30}-\frac{1}{5}a^{29}+\frac{1}{5}a^{28}-\frac{2}{5}a^{27}+\frac{2}{5}a^{26}-\frac{2}{5}a^{24}+\frac{1}{5}a^{22}-\frac{2}{5}a^{21}+\frac{1}{5}a^{19}-\frac{1}{5}a^{18}+\frac{2}{5}a^{16}-\frac{2}{5}a^{15}+\frac{2}{5}a^{14}+\frac{2}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{40}+\frac{1}{5}a^{38}-\frac{1}{5}a^{37}+\frac{1}{5}a^{36}-\frac{2}{5}a^{35}-\frac{2}{5}a^{34}+\frac{2}{5}a^{32}+\frac{1}{5}a^{30}-\frac{2}{5}a^{29}+\frac{1}{5}a^{28}+\frac{1}{5}a^{27}+\frac{1}{5}a^{26}-\frac{2}{5}a^{25}-\frac{1}{5}a^{24}+\frac{1}{5}a^{23}+\frac{1}{5}a^{22}-\frac{1}{5}a^{21}+\frac{1}{5}a^{20}+\frac{2}{5}a^{19}+\frac{2}{5}a^{18}+\frac{2}{5}a^{17}-\frac{1}{5}a^{16}+\frac{1}{5}a^{15}-\frac{2}{5}a^{14}-\frac{1}{5}a^{13}-\frac{1}{5}a^{12}+\frac{1}{5}a^{11}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{41}+\frac{2}{5}a^{38}+\frac{1}{5}a^{37}-\frac{1}{5}a^{36}-\frac{1}{5}a^{35}-\frac{1}{5}a^{34}+\frac{2}{5}a^{33}-\frac{1}{5}a^{31}-\frac{1}{5}a^{30}+\frac{2}{5}a^{29}-\frac{2}{5}a^{27}+\frac{1}{5}a^{26}-\frac{1}{5}a^{25}-\frac{2}{5}a^{24}+\frac{1}{5}a^{23}-\frac{2}{5}a^{22}-\frac{2}{5}a^{21}+\frac{2}{5}a^{20}+\frac{1}{5}a^{19}-\frac{2}{5}a^{18}-\frac{1}{5}a^{17}-\frac{1}{5}a^{16}+\frac{2}{5}a^{14}+\frac{2}{5}a^{13}-\frac{2}{5}a^{12}-\frac{1}{5}a^{11}-\frac{1}{5}a^{9}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{42}+\frac{2}{5}a^{38}-\frac{1}{5}a^{37}+\frac{1}{5}a^{36}+\frac{1}{5}a^{35}-\frac{1}{5}a^{32}-\frac{1}{5}a^{30}+\frac{2}{5}a^{29}+\frac{1}{5}a^{28}-\frac{2}{5}a^{25}-\frac{2}{5}a^{23}+\frac{1}{5}a^{22}+\frac{1}{5}a^{21}+\frac{1}{5}a^{20}+\frac{1}{5}a^{19}+\frac{1}{5}a^{18}-\frac{1}{5}a^{17}+\frac{1}{5}a^{16}+\frac{1}{5}a^{15}-\frac{2}{5}a^{14}-\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{2}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{3954365}a^{43}+\frac{139139}{3954365}a^{42}+\frac{11330}{790873}a^{41}+\frac{3541}{3954365}a^{40}-\frac{279824}{3954365}a^{39}+\frac{1812486}{3954365}a^{38}+\frac{1814426}{3954365}a^{37}-\frac{1484408}{3954365}a^{36}-\frac{828827}{3954365}a^{35}+\frac{641927}{3954365}a^{34}-\frac{208161}{3954365}a^{33}+\frac{1322943}{3954365}a^{32}+\frac{605952}{3954365}a^{31}-\frac{316670}{790873}a^{30}+\frac{8613}{3954365}a^{29}-\frac{104056}{3954365}a^{28}+\frac{1224078}{3954365}a^{27}+\frac{362882}{3954365}a^{26}-\frac{163405}{790873}a^{25}+\frac{319669}{3954365}a^{24}-\frac{1829576}{3954365}a^{23}-\frac{362348}{790873}a^{22}+\frac{633621}{3954365}a^{21}+\frac{1506991}{3954365}a^{20}+\frac{973826}{3954365}a^{19}-\frac{303099}{3954365}a^{18}-\frac{840401}{3954365}a^{17}-\frac{1836818}{3954365}a^{16}+\frac{307124}{790873}a^{15}-\frac{1454053}{3954365}a^{14}-\frac{1973834}{3954365}a^{13}-\frac{389537}{3954365}a^{12}-\frac{1697687}{3954365}a^{11}-\frac{744176}{3954365}a^{10}-\frac{1245754}{3954365}a^{9}-\frac{208297}{3954365}a^{8}-\frac{251837}{790873}a^{7}-\frac{136378}{790873}a^{6}-\frac{1079719}{3954365}a^{5}-\frac{1472134}{3954365}a^{4}-\frac{935654}{3954365}a^{3}-\frac{1696366}{3954365}a^{2}-\frac{344266}{3954365}a+\frac{1276769}{3954365}$, $\frac{1}{80\!\cdots\!35}a^{44}+\frac{17\!\cdots\!23}{16\!\cdots\!27}a^{43}-\frac{66\!\cdots\!49}{80\!\cdots\!35}a^{42}-\frac{69\!\cdots\!61}{80\!\cdots\!35}a^{41}+\frac{80\!\cdots\!36}{16\!\cdots\!27}a^{40}-\frac{69\!\cdots\!93}{80\!\cdots\!35}a^{39}+\frac{56\!\cdots\!76}{16\!\cdots\!27}a^{38}+\frac{44\!\cdots\!66}{80\!\cdots\!35}a^{37}-\frac{42\!\cdots\!78}{80\!\cdots\!35}a^{36}-\frac{25\!\cdots\!87}{80\!\cdots\!35}a^{35}+\frac{14\!\cdots\!97}{80\!\cdots\!35}a^{34}-\frac{85\!\cdots\!22}{80\!\cdots\!35}a^{33}-\frac{23\!\cdots\!21}{80\!\cdots\!35}a^{32}+\frac{51\!\cdots\!89}{80\!\cdots\!35}a^{31}+\frac{35\!\cdots\!41}{80\!\cdots\!35}a^{30}+\frac{32\!\cdots\!61}{80\!\cdots\!35}a^{29}-\frac{11\!\cdots\!38}{25\!\cdots\!85}a^{28}-\frac{18\!\cdots\!38}{80\!\cdots\!35}a^{27}-\frac{41\!\cdots\!97}{80\!\cdots\!35}a^{26}+\frac{18\!\cdots\!91}{80\!\cdots\!35}a^{25}-\frac{79\!\cdots\!01}{80\!\cdots\!35}a^{24}-\frac{24\!\cdots\!19}{80\!\cdots\!35}a^{23}+\frac{50\!\cdots\!81}{16\!\cdots\!27}a^{22}+\frac{35\!\cdots\!98}{16\!\cdots\!27}a^{21}-\frac{68\!\cdots\!77}{80\!\cdots\!35}a^{20}+\frac{23\!\cdots\!23}{80\!\cdots\!35}a^{19}+\frac{14\!\cdots\!07}{80\!\cdots\!35}a^{18}-\frac{25\!\cdots\!48}{80\!\cdots\!35}a^{17}-\frac{19\!\cdots\!77}{80\!\cdots\!35}a^{16}-\frac{16\!\cdots\!03}{80\!\cdots\!35}a^{15}-\frac{34\!\cdots\!61}{80\!\cdots\!35}a^{14}+\frac{71\!\cdots\!44}{16\!\cdots\!27}a^{13}-\frac{33\!\cdots\!02}{80\!\cdots\!35}a^{12}-\frac{15\!\cdots\!63}{80\!\cdots\!35}a^{11}-\frac{19\!\cdots\!19}{80\!\cdots\!35}a^{10}+\frac{68\!\cdots\!27}{16\!\cdots\!27}a^{9}+\frac{19\!\cdots\!21}{80\!\cdots\!35}a^{8}-\frac{30\!\cdots\!51}{80\!\cdots\!35}a^{7}-\frac{19\!\cdots\!39}{80\!\cdots\!35}a^{6}-\frac{94\!\cdots\!24}{80\!\cdots\!35}a^{5}-\frac{59\!\cdots\!17}{16\!\cdots\!27}a^{4}-\frac{79\!\cdots\!29}{16\!\cdots\!27}a^{3}-\frac{25\!\cdots\!54}{80\!\cdots\!35}a^{2}-\frac{29\!\cdots\!42}{16\!\cdots\!27}a+\frac{24\!\cdots\!84}{12\!\cdots\!65}$ 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 - 12*x^44 - 38*x^43 + 936*x^42 - 488*x^41 - 32994*x^40 + 61155*x^39 + 695151*x^38 - 1826452*x^37 - 9745559*x^36 + 31398650*x^35 + 95576968*x^34 - 363031278*x^33 - 668271620*x^32 + 3006554151*x^31 + 3299628259*x^30 - 18405469866*x^29 - 10819187452*x^28 + 84606290574*x^27 + 17798672756*x^26 - 293624042239*x^25 + 25779977473*x^24 + 766879229695*x^23 - 259704306954*x^22 - 1488735953755*x^21 + 843265985246*x^20 + 2094958274346*x^19 - 1672596038928*x^18 - 2036992655261*x^17 + 2211683013511*x^16 + 1229593352332*x^15 - 1960884055901*x^14 - 305227614428*x^13 + 1128154934100*x^12 - 126915280282*x^11 - 391023252143*x^10 + 124192084723*x^9 + 69369604379*x^8 - 37452923715*x^7 - 3334757683*x^6 + 4610823573*x^5 - 413774775*x^4 - 188088413*x^3 + 35065204*x^2 - 872335*x + 619)
 
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 - 12*x^44 - 38*x^43 + 936*x^42 - 488*x^41 - 32994*x^40 + 61155*x^39 + 695151*x^38 - 1826452*x^37 - 9745559*x^36 + 31398650*x^35 + 95576968*x^34 - 363031278*x^33 - 668271620*x^32 + 3006554151*x^31 + 3299628259*x^30 - 18405469866*x^29 - 10819187452*x^28 + 84606290574*x^27 + 17798672756*x^26 - 293624042239*x^25 + 25779977473*x^24 + 766879229695*x^23 - 259704306954*x^22 - 1488735953755*x^21 + 843265985246*x^20 + 2094958274346*x^19 - 1672596038928*x^18 - 2036992655261*x^17 + 2211683013511*x^16 + 1229593352332*x^15 - 1960884055901*x^14 - 305227614428*x^13 + 1128154934100*x^12 - 126915280282*x^11 - 391023252143*x^10 + 124192084723*x^9 + 69369604379*x^8 - 37452923715*x^7 - 3334757683*x^6 + 4610823573*x^5 - 413774775*x^4 - 188088413*x^3 + 35065204*x^2 - 872335*x + 619, 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 - 12*x^44 - 38*x^43 + 936*x^42 - 488*x^41 - 32994*x^40 + 61155*x^39 + 695151*x^38 - 1826452*x^37 - 9745559*x^36 + 31398650*x^35 + 95576968*x^34 - 363031278*x^33 - 668271620*x^32 + 3006554151*x^31 + 3299628259*x^30 - 18405469866*x^29 - 10819187452*x^28 + 84606290574*x^27 + 17798672756*x^26 - 293624042239*x^25 + 25779977473*x^24 + 766879229695*x^23 - 259704306954*x^22 - 1488735953755*x^21 + 843265985246*x^20 + 2094958274346*x^19 - 1672596038928*x^18 - 2036992655261*x^17 + 2211683013511*x^16 + 1229593352332*x^15 - 1960884055901*x^14 - 305227614428*x^13 + 1128154934100*x^12 - 126915280282*x^11 - 391023252143*x^10 + 124192084723*x^9 + 69369604379*x^8 - 37452923715*x^7 - 3334757683*x^6 + 4610823573*x^5 - 413774775*x^4 - 188088413*x^3 + 35065204*x^2 - 872335*x + 619);
 
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 - 12*x^44 - 38*x^43 + 936*x^42 - 488*x^41 - 32994*x^40 + 61155*x^39 + 695151*x^38 - 1826452*x^37 - 9745559*x^36 + 31398650*x^35 + 95576968*x^34 - 363031278*x^33 - 668271620*x^32 + 3006554151*x^31 + 3299628259*x^30 - 18405469866*x^29 - 10819187452*x^28 + 84606290574*x^27 + 17798672756*x^26 - 293624042239*x^25 + 25779977473*x^24 + 766879229695*x^23 - 259704306954*x^22 - 1488735953755*x^21 + 843265985246*x^20 + 2094958274346*x^19 - 1672596038928*x^18 - 2036992655261*x^17 + 2211683013511*x^16 + 1229593352332*x^15 - 1960884055901*x^14 - 305227614428*x^13 + 1128154934100*x^12 - 126915280282*x^11 - 391023252143*x^10 + 124192084723*x^9 + 69369604379*x^8 - 37452923715*x^7 - 3334757683*x^6 + 4610823573*x^5 - 413774775*x^4 - 188088413*x^3 + 35065204*x^2 - 872335*x + 619);
 
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

3.3.162409.1, 3.3.961.1, 3.3.169.1, 3.3.162409.2, 5.5.923521.1, 9.9.4283810754983929.1, 15.15.104351149323786133540261771563529.1, \(\Q(\zeta_{31})^+\), 15.15.108586003458674436566349398089.1, 15.15.104351149323786133540261771563529.2

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 $15^{3}$ $15^{3}$ ${\href{/padicField/5.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ R $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ R ${\href{/padicField/37.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ ${\href{/padicField/47.5.0.1}{5} }^{9}$ $15^{3}$ $15^{3}$

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
\(13\) Copy content Toggle raw display Deg $45$$3$$15$$30$
\(31\) Copy content Toggle raw display 31.15.14.1$x^{15} + 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$
31.15.14.1$x^{15} + 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$
31.15.14.1$x^{15} + 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$