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Properties

Label 36.0.131...081.1
Degree $36$
Signature $[0, 18]$
Discriminant $1.311\times 10^{51}$
Root discriminant \(26.30\)
Ramified primes $3,7$
Class number $7$ (GRH)
Class group [7] (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 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1)
 
gp: K = bnfinit(y^36 - y^33 + y^27 - y^24 + y^18 - y^12 + y^9 - y^3 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1)
 

\( x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 \) 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:   \(1310656710125779295611091389185381649163216325999081\) \(\medspace = 3^{54}\cdot 7^{30}\) 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:  \(26.30\)
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:  $3^{3/2}7^{5/6}\approx 26.298455832887633$
Ramified primes:   \(3\), \(7\) 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:  \(63=3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{63}(1,·)$, $\chi_{63}(2,·)$, $\chi_{63}(4,·)$, $\chi_{63}(5,·)$, $\chi_{63}(8,·)$, $\chi_{63}(10,·)$, $\chi_{63}(11,·)$, $\chi_{63}(13,·)$, $\chi_{63}(16,·)$, $\chi_{63}(17,·)$, $\chi_{63}(19,·)$, $\chi_{63}(20,·)$, $\chi_{63}(22,·)$, $\chi_{63}(23,·)$, $\chi_{63}(25,·)$, $\chi_{63}(26,·)$, $\chi_{63}(29,·)$, $\chi_{63}(31,·)$, $\chi_{63}(32,·)$, $\chi_{63}(34,·)$, $\chi_{63}(37,·)$, $\chi_{63}(38,·)$, $\chi_{63}(40,·)$, $\chi_{63}(41,·)$, $\chi_{63}(43,·)$, $\chi_{63}(44,·)$, $\chi_{63}(46,·)$, $\chi_{63}(47,·)$, $\chi_{63}(50,·)$, $\chi_{63}(52,·)$, $\chi_{63}(53,·)$, $\chi_{63}(55,·)$, $\chi_{63}(58,·)$, $\chi_{63}(59,·)$, $\chi_{63}(61,·)$, $\chi_{63}(62,·)$$\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

$C_{7}$, which has order $7$ (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:   \( -a \)  (order $126$) 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:   $a^{9}+1$, $a^{33}+a^{24}+a^{15}+a^{6}$, $a^{6}-1$, $a^{12}-1$, $a^{33}+a^{30}+a^{12}$, $a^{7}+1$, $a^{35}+a^{7}$, $a^{2}-1$, $a-1$, $a^{4}-1$, $a^{34}-a^{31}+a^{25}-a^{22}-a^{20}+a^{16}-a^{10}+a^{7}-a$, $a^{4}+1$, $a^{29}+a^{19}+a^{8}$, $a^{32}+a^{11}-a^{3}$, $a^{13}-1$, $a^{34}-a^{28}+a^{25}-a^{19}-a^{18}+a^{16}+a^{13}-a^{10}+a^{4}-a$, $a^{25}+a^{15}$ 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:  \( 408184875586.38837 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 408184875586.38837 \cdot 7}{126\cdot\sqrt{1310656710125779295611091389185381649163216325999081}}\cr\approx \mathstrut & 0.145907479932327 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 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 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 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 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 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 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 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 $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{9})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\zeta_{7})\), 6.0.2250423.1, 6.0.110270727.1, 6.0.110270727.2, 6.0.64827.1, \(\Q(\zeta_{21})^+\), \(\Q(\zeta_{9})\), 6.6.6751269.1, 6.0.47258883.1, 6.6.330812181.1, 6.0.47258883.2, 6.6.330812181.2, 9.9.62523502209.1, \(\Q(\zeta_{21})\), 12.0.45579633110361.1, 12.0.109436699097976761.2, 12.0.109436699097976761.1, 18.0.1340851596668237962730583.1, 18.0.105548084868928352751387.1, \(\Q(\zeta_{63})^+\)

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 ${\href{/padicField/2.6.0.1}{6} }^{6}$ R ${\href{/padicField/5.6.0.1}{6} }^{6}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{12}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.3.0.1}{3} }^{12}$ ${\href{/padicField/47.6.0.1}{6} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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
\(3\) Copy content Toggle raw display Deg $36$$6$$6$$54$
\(7\) Copy content Toggle raw display 7.18.15.5$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$$6$$3$$15$$C_6 \times C_3$$[\ ]_{6}^{3}$
7.18.15.5$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$$6$$3$$15$$C_6 \times C_3$$[\ ]_{6}^{3}$