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Properties

Label 36.0.900...616.1
Degree $36$
Signature $[0, 18]$
Discriminant $9.007\times 10^{61}$
Root discriminant \(52.60\)
Ramified primes $2,3,7$
Class number $20384$ (GRH)
Class group [2, 28, 364] (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 36*x^34 + 594*x^32 + 5953*x^30 + 40485*x^28 + 197721*x^26 + 715780*x^24 + 1954770*x^22 + 4059891*x^20 + 6417344*x^18 + 7674462*x^16 + 6854571*x^14 + 4475587*x^12 + 2066547*x^10 + 640764*x^8 + 122466*x^6 + 12276*x^4 + 432*x^2 + 1)
 
gp: K = bnfinit(y^36 + 36*y^34 + 594*y^32 + 5953*y^30 + 40485*y^28 + 197721*y^26 + 715780*y^24 + 1954770*y^22 + 4059891*y^20 + 6417344*y^18 + 7674462*y^16 + 6854571*y^14 + 4475587*y^12 + 2066547*y^10 + 640764*y^8 + 122466*y^6 + 12276*y^4 + 432*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 36*x^34 + 594*x^32 + 5953*x^30 + 40485*x^28 + 197721*x^26 + 715780*x^24 + 1954770*x^22 + 4059891*x^20 + 6417344*x^18 + 7674462*x^16 + 6854571*x^14 + 4475587*x^12 + 2066547*x^10 + 640764*x^8 + 122466*x^6 + 12276*x^4 + 432*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 36*x^34 + 594*x^32 + 5953*x^30 + 40485*x^28 + 197721*x^26 + 715780*x^24 + 1954770*x^22 + 4059891*x^20 + 6417344*x^18 + 7674462*x^16 + 6854571*x^14 + 4475587*x^12 + 2066547*x^10 + 640764*x^8 + 122466*x^6 + 12276*x^4 + 432*x^2 + 1)
 

\( x^{36} + 36 x^{34} + 594 x^{32} + 5953 x^{30} + 40485 x^{28} + 197721 x^{26} + 715780 x^{24} + 1954770 x^{22} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(90067643300370785938616861622694756230952958181429238736879616\) \(\medspace = 2^{36}\cdot 3^{54}\cdot 7^{30}\) 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:  \(52.60\)
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:  $2\cdot 3^{3/2}7^{5/6}\approx 52.596911665775266$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(131,·)$, $\chi_{252}(5,·)$, $\chi_{252}(143,·)$, $\chi_{252}(17,·)$, $\chi_{252}(151,·)$, $\chi_{252}(25,·)$, $\chi_{252}(163,·)$, $\chi_{252}(37,·)$, $\chi_{252}(167,·)$, $\chi_{252}(41,·)$, $\chi_{252}(43,·)$, $\chi_{252}(173,·)$, $\chi_{252}(47,·)$, $\chi_{252}(185,·)$, $\chi_{252}(59,·)$, $\chi_{252}(169,·)$, $\chi_{252}(193,·)$, $\chi_{252}(67,·)$, $\chi_{252}(205,·)$, $\chi_{252}(79,·)$, $\chi_{252}(209,·)$, $\chi_{252}(83,·)$, $\chi_{252}(85,·)$, $\chi_{252}(215,·)$, $\chi_{252}(89,·)$, $\chi_{252}(227,·)$, $\chi_{252}(101,·)$, $\chi_{252}(235,·)$, $\chi_{252}(109,·)$, $\chi_{252}(211,·)$, $\chi_{252}(247,·)$, $\chi_{252}(121,·)$, $\chi_{252}(251,·)$, $\chi_{252}(125,·)$, $\chi_{252}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{28}\times C_{364}$, which has order $20384$ (assuming GRH)

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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( a^{21} + 21 a^{19} + 189 a^{17} + 952 a^{15} + 2940 a^{13} + 5733 a^{11} + 7007 a^{9} + 5148 a^{7} + 2079 a^{5} + 385 a^{3} + 21 a \)  (order $4$) 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:   $a^{9}+9a^{7}+27a^{5}+30a^{3}+9a$, $a^{27}+27a^{25}+324a^{23}+2277a^{21}+10395a^{19}+32319a^{17}+69768a^{15}+104652a^{13}+107406a^{11}+72930a^{9}+30888a^{7}+7371a^{5}+819a^{3}+27a$, $a^{7}+7a^{5}+14a^{3}+7a$, $a^{21}+21a^{19}+189a^{17}+952a^{15}+2940a^{13}+5733a^{11}+7007a^{9}+5147a^{7}+2072a^{5}+371a^{3}+14a$, $a^{31}+31a^{29}+434a^{27}+3627a^{25}+20150a^{23}+78430a^{21}+219604a^{19}+447051a^{17}+660858a^{15}+700910a^{13}+520677a^{11}+260349a^{9}+82256a^{7}+14833a^{5}+1295a^{3}+42a$, $a^{17}+17a^{15}+119a^{13}+442a^{11}+935a^{9}+1122a^{7}+714a^{5}+204a^{3}+17a$, $a^{23}+23a^{21}+231a^{19}+1330a^{17}+4844a^{15}+11613a^{13}+18473a^{11}+19162a^{9}+12375a^{7}+4543a^{5}+791a^{3}+42a$, $a^{8}+8a^{6}+20a^{4}+16a^{2}+2$, $a$, $a^{23}+23a^{21}+230a^{19}+1311a^{17}+4692a^{15}+10948a^{13}+16744a^{11}+16445a^{9}+9867a^{7}+3289a^{5}+506a^{3}+23a$, $a^{35}+35a^{33}+559a^{31}+5394a^{29}+35091a^{27}+162629a^{25}+553124a^{23}+1401323a^{21}+2656312a^{19}+3750826a^{17}+3892269a^{15}+2895474a^{13}+1481194a^{11}+484954a^{9}+88029a^{7}+5637a^{5}-320a^{3}-15a$, $a^{25}+25a^{23}+275a^{21}+1750a^{19}+7126a^{17}+19397a^{15}+35819a^{13}+44642a^{11}+36685a^{9}+18997a^{7}+5719a^{5}+854a^{3}+42a$, $a^{29}+29a^{27}+377a^{25}+2900a^{23}+14674a^{21}+51359a^{19}+127281a^{17}+224808a^{15}+281010a^{13}+243542a^{11}+140998a^{9}+51272a^{7}+10556a^{5}+1015a^{3}+29a$, $a^{34}+34a^{32}+527a^{30}+4930a^{28}+31059a^{26}+139230a^{24}+457470a^{22}+1118260a^{20}+2042975a^{18}+2778446a^{16}+2778446a^{14}+1998724a^{12}+999362a^{10}+329459a^{8}+65884a^{6}+6916a^{4}+273a^{2}$, $a^{31}+31a^{29}+434a^{27}+3627a^{25}+20150a^{23}+78430a^{21}+219603a^{19}+447032a^{17}+660706a^{15}+700244a^{13}+518934a^{11}+257556a^{9}+79548a^{7}+13319a^{5}+859a^{3}-5a$, $a^{11}+11a^{9}+44a^{7}+77a^{5}+55a^{3}+11a$, $a^{19}+19a^{17}+152a^{15}+665a^{13}+1729a^{11}+2717a^{9}+2508a^{7}+1254a^{5}+285a^{3}+19a$ Copy content Toggle raw display (assuming GRH)
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:  \( 816369751172.7767 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 816369751172.7767 \cdot 20384}{4\cdot\sqrt{90067643300370785938616861622694756230952958181429238736879616}}\cr\approx \mathstrut & 0.102110300592289 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 + 36*x^34 + 594*x^32 + 5953*x^30 + 40485*x^28 + 197721*x^26 + 715780*x^24 + 1954770*x^22 + 4059891*x^20 + 6417344*x^18 + 7674462*x^16 + 6854571*x^14 + 4475587*x^12 + 2066547*x^10 + 640764*x^8 + 122466*x^6 + 12276*x^4 + 432*x^2 + 1)
 
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^36 + 36*x^34 + 594*x^32 + 5953*x^30 + 40485*x^28 + 197721*x^26 + 715780*x^24 + 1954770*x^22 + 4059891*x^20 + 6417344*x^18 + 7674462*x^16 + 6854571*x^14 + 4475587*x^12 + 2066547*x^10 + 640764*x^8 + 122466*x^6 + 12276*x^4 + 432*x^2 + 1, 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^36 + 36*x^34 + 594*x^32 + 5953*x^30 + 40485*x^28 + 197721*x^26 + 715780*x^24 + 1954770*x^22 + 4059891*x^20 + 6417344*x^18 + 7674462*x^16 + 6854571*x^14 + 4475587*x^12 + 2066547*x^10 + 640764*x^8 + 122466*x^6 + 12276*x^4 + 432*x^2 + 1);
 
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^36 + 36*x^34 + 594*x^32 + 5953*x^30 + 40485*x^28 + 197721*x^26 + 715780*x^24 + 1954770*x^22 + 4059891*x^20 + 6417344*x^18 + 7674462*x^16 + 6854571*x^14 + 4475587*x^12 + 2066547*x^10 + 640764*x^8 + 122466*x^6 + 12276*x^4 + 432*x^2 + 1);
 
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

$C_6^2$ (as 36T4):

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 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-21}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(i, \sqrt{21})\), 6.0.419904.1, 6.0.1008189504.2, 6.0.153664.1, 6.0.1008189504.1, 6.6.6751269.1, 6.0.432081216.1, 6.6.330812181.1, 6.0.21171979584.1, \(\Q(\zeta_{21})^+\), 6.0.29042496.1, 6.6.330812181.2, 6.0.21171979584.2, 9.9.62523502209.1, 12.0.186694177220038656.1, 12.0.448252719505312813056.2, 12.0.843466573910016.1, 12.0.448252719505312813056.1, 18.0.1024770265180753855691096064.1, \(\Q(\zeta_{63})^+\), 18.0.9490397425838961457555240648704.1

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 R R ${\href{/padicField/5.3.0.1}{3} }^{12}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{6}$ ${\href{/padicField/17.3.0.1}{3} }^{12}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{12}$ ${\href{/padicField/41.3.0.1}{3} }^{12}$ ${\href{/padicField/43.6.0.1}{6} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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
\(2\) Copy content Toggle raw display 2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
\(3\) Copy content Toggle raw display Deg $36$$6$$6$$54$
\(7\) Copy content Toggle raw display Deg $36$$6$$6$$30$