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Properties

Label 36.0.774...632.1
Degree $36$
Signature $[0, 18]$
Discriminant $7.746\times 10^{61}$
Root discriminant \(52.38\)
Ramified primes $2,3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{18}:C_6$ (as 36T185)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 3)
 
gp: K = bnfinit(y^36 + 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 3);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 3)
 

\( x^{36} + 3 \) 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:   \(77455827645541172243429237514462094454119707909588182611525632\) \(\medspace = 2^{36}\cdot 3^{107}\) 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:  \(52.38\)
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:  $2^{3/2}3^{331/108}\approx 82.00366889518594$
Ramified primes:   \(2\), \(3\) 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:  \(\Q(\sqrt{3}) \)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Trivial group, which has order $1$ (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:   \( \frac{1}{2} a^{18} + \frac{1}{2} \)  (order $6$) 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:   $\frac{1}{2}a^{27}+\frac{1}{2}a^{18}+\frac{1}{2}a^{9}-\frac{1}{2}$, $\frac{1}{2}a^{18}-a^{12}+a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{24}-a^{12}-\frac{1}{2}a^{6}+1$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{30}+\frac{1}{2}a^{27}-\frac{1}{2}a^{24}+\frac{1}{2}a^{21}-\frac{1}{2}a^{18}+\frac{1}{2}a^{15}-\frac{1}{2}a^{12}+\frac{1}{2}a^{9}-\frac{1}{2}a^{6}+\frac{1}{2}a^{3}+\frac{1}{2}$, $\frac{1}{2}a^{34}+\frac{1}{2}a^{33}-\frac{1}{2}a^{31}-\frac{1}{2}a^{30}+\frac{1}{2}a^{28}+\frac{1}{2}a^{27}-\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-a^{17}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+a^{14}+\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-a^{11}-\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+a^{8}+\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-a^{5}-\frac{1}{2}a^{4}+\frac{1}{2}a^{3}+a^{2}+\frac{1}{2}a+\frac{1}{2}$, $\frac{1}{2}a^{35}+\frac{1}{2}a^{34}+\frac{1}{2}a^{33}+\frac{1}{2}a^{32}+\frac{1}{2}a^{31}+\frac{1}{2}a^{30}+\frac{1}{2}a^{29}+\frac{1}{2}a^{28}+\frac{1}{2}a^{27}+\frac{1}{2}a^{26}+\frac{1}{2}a^{25}+\frac{1}{2}a^{24}+\frac{1}{2}a^{23}+\frac{1}{2}a^{22}+\frac{1}{2}a^{21}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{11}+\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+\frac{1}{2}a^{5}+\frac{1}{2}a^{4}+\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{32}+\frac{1}{2}a^{30}-\frac{1}{2}a^{28}+\frac{1}{2}a^{26}-\frac{1}{2}a^{24}+\frac{1}{2}a^{22}-\frac{1}{2}a^{20}+\frac{1}{2}a^{18}-\frac{1}{2}a^{16}+\frac{1}{2}a^{14}-\frac{1}{2}a^{12}+\frac{1}{2}a^{10}-\frac{1}{2}a^{8}+\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}+\frac{1}{2}$, $\frac{1}{2}a^{32}+\frac{1}{2}a^{30}-\frac{1}{2}a^{26}-\frac{1}{2}a^{24}+\frac{1}{2}a^{20}-a^{16}-\frac{1}{2}a^{14}+\frac{1}{2}a^{12}+a^{10}+\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-a^{4}+\frac{1}{2}a^{2}+1$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{32}-\frac{1}{2}a^{30}+\frac{1}{2}a^{28}+\frac{1}{2}a^{26}-\frac{1}{2}a^{24}-\frac{1}{2}a^{22}+\frac{1}{2}a^{20}+\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}+\frac{1}{2}a^{12}+\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}+\frac{1}{2}a^{4}+\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{34}+\frac{1}{2}a^{32}-\frac{1}{2}a^{28}-\frac{1}{2}a^{26}+\frac{1}{2}a^{22}+\frac{1}{2}a^{20}+\frac{1}{2}a^{18}+\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-a^{12}-\frac{1}{2}a^{10}+\frac{1}{2}a^{8}+a^{6}+\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{27}+\frac{1}{2}a^{21}-a^{15}+\frac{1}{2}a^{12}+\frac{1}{2}a^{9}-a^{6}+\frac{1}{2}a^{3}+1$, $\frac{1}{2}a^{34}+a^{32}+a^{30}+\frac{1}{2}a^{28}-\frac{1}{2}a^{24}-a^{22}-a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{16}+\frac{1}{2}a^{10}+a^{8}+\frac{3}{2}a^{6}+2a^{4}+a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{24}-\frac{1}{2}a^{18}+a^{16}-\frac{1}{2}a^{8}+\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{11}{2}a^{35}-a^{34}-5a^{33}+\frac{5}{2}a^{32}+\frac{5}{2}a^{31}-\frac{11}{2}a^{30}+\frac{3}{2}a^{29}+\frac{13}{2}a^{28}-3a^{27}-6a^{26}+\frac{9}{2}a^{25}+2a^{24}-\frac{13}{2}a^{23}+4a^{22}+7a^{21}-6a^{20}-6a^{19}+7a^{18}+\frac{1}{2}a^{17}-7a^{16}+7a^{15}+\frac{13}{2}a^{14}-\frac{19}{2}a^{13}-\frac{9}{2}a^{12}+\frac{19}{2}a^{11}-\frac{3}{2}a^{10}-7a^{9}+10a^{8}+\frac{9}{2}a^{7}-13a^{6}-\frac{3}{2}a^{5}+12a^{4}-4a^{3}-7a^{2}+13a+1$, $\frac{3}{2}a^{35}-\frac{1}{2}a^{34}-\frac{3}{2}a^{33}+4a^{32}-\frac{9}{2}a^{31}+4a^{30}-\frac{11}{2}a^{29}+7a^{28}-\frac{13}{2}a^{27}+\frac{9}{2}a^{26}-\frac{7}{2}a^{25}+4a^{24}-\frac{5}{2}a^{23}-\frac{3}{2}a^{22}+3a^{21}-\frac{9}{2}a^{20}+\frac{9}{2}a^{19}-\frac{17}{2}a^{18}+\frac{19}{2}a^{17}-\frac{17}{2}a^{16}+\frac{13}{2}a^{15}-8a^{14}+\frac{15}{2}a^{13}-5a^{12}-\frac{1}{2}a^{11}-\frac{5}{2}a^{9}+\frac{9}{2}a^{8}-\frac{23}{2}a^{7}+12a^{6}-\frac{21}{2}a^{5}+\frac{21}{2}a^{4}-14a^{3}+\frac{25}{2}a^{2}-\frac{17}{2}a+\frac{5}{2}$, $\frac{3}{2}a^{35}-a^{34}+\frac{1}{2}a^{33}+\frac{7}{2}a^{32}-3a^{31}+2a^{30}-\frac{1}{2}a^{29}-4a^{28}+a^{26}-\frac{11}{2}a^{25}+2a^{24}-a^{23}-\frac{3}{2}a^{22}+\frac{7}{2}a^{21}-3a^{19}+4a^{18}-\frac{7}{2}a^{17}-a^{16}+\frac{13}{2}a^{15}-\frac{3}{2}a^{14}+6a^{12}-\frac{3}{2}a^{11}+2a^{10}+5a^{9}-7a^{8}+\frac{3}{2}a^{7}-7a^{5}+\frac{5}{2}a^{4}+\frac{7}{2}a^{3}-11a^{2}+5a-1$, $\frac{1}{2}a^{35}+\frac{3}{2}a^{34}+2a^{33}+a^{32}+a^{31}-2a^{30}-\frac{3}{2}a^{27}-4a^{26}-\frac{3}{2}a^{25}+\frac{1}{2}a^{24}+a^{23}-a^{22}+a^{21}+2a^{20}+\frac{9}{2}a^{19}+\frac{5}{2}a^{18}-\frac{3}{2}a^{17}-\frac{1}{2}a^{16}+2a^{15}-2a^{14}-4a^{13}-3a^{12}-3a^{11}-\frac{1}{2}a^{9}+\frac{3}{2}a^{7}+\frac{9}{2}a^{6}+3a^{5}+3a^{4}+2a^{3}+3a^{2}-\frac{9}{2}a-\frac{1}{2}$ 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:  \( 146190247413682400 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 146190247413682400 \cdot 1}{6\cdot\sqrt{77455827645541172243429237514462094454119707909588182611525632}}\cr\approx \mathstrut & 0.644878253131210 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 + 3)
 
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 + 3, 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 + 3);
 
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 + 3);
 
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

$D_{18}:C_6$ (as 36T185):

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 solvable group of order 216
The 31 conjugacy class representatives for $D_{18}:C_6$
Character table for $D_{18}:C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 4.0.432.1, 6.0.177147.2, 9.1.2541865828329.2 x3, 12.0.385610460475392.3, 18.0.19383245667680019896796723.2

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]
 

Sibling fields

Degree 36 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.12.0.1}{12} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}$ $18{,}\,{\href{/padicField/7.9.0.1}{9} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}$ $18^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{9}$ $18{,}\,{\href{/padicField/19.9.0.1}{9} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.12.0.1}{12} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{3}$ $18{,}\,{\href{/padicField/31.9.0.1}{9} }^{2}$ ${\href{/padicField/37.9.0.1}{9} }^{4}$ ${\href{/padicField/41.12.0.1}{12} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{3}$ $18{,}\,{\href{/padicField/43.9.0.1}{9} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{9}$ ${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}$

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 2.4.4.4$x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.12.12.18$x^{12} - 10 x^{11} + 72 x^{10} - 332 x^{9} + 1316 x^{8} - 4160 x^{7} + 12128 x^{6} - 27904 x^{5} + 53744 x^{4} - 69600 x^{3} + 71680 x^{2} - 41536 x + 38848$$2$$6$$12$$D_4 \times C_3$$[2, 2]^{6}$
2.12.12.18$x^{12} - 10 x^{11} + 72 x^{10} - 332 x^{9} + 1316 x^{8} - 4160 x^{7} + 12128 x^{6} - 27904 x^{5} + 53744 x^{4} - 69600 x^{3} + 71680 x^{2} - 41536 x + 38848$$2$$6$$12$$D_4 \times C_3$$[2, 2]^{6}$
\(3\) Copy content Toggle raw display Deg $36$$36$$1$$107$