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Properties

Label 12.0.800...125.1
Degree $12$
Signature $[0, 6]$
Discriminant $8.007\times 10^{23}$
Root discriminant \(98.16\)
Ramified primes $5,17,19$
Class number $2834$ (GRH)
Class group [2834] (GRH)
Galois group $C_{12}$ (as 12T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 37*x^10 + 21*x^9 + 997*x^8 + 268*x^7 + 16408*x^6 + 7328*x^5 + 208512*x^4 - 219292*x^3 + 2113070*x^2 - 1950049*x + 6498881)
 
gp: K = bnfinit(y^12 - y^11 + 37*y^10 + 21*y^9 + 997*y^8 + 268*y^7 + 16408*y^6 + 7328*y^5 + 208512*y^4 - 219292*y^3 + 2113070*y^2 - 1950049*y + 6498881, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 37*x^10 + 21*x^9 + 997*x^8 + 268*x^7 + 16408*x^6 + 7328*x^5 + 208512*x^4 - 219292*x^3 + 2113070*x^2 - 1950049*x + 6498881);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 37*x^10 + 21*x^9 + 997*x^8 + 268*x^7 + 16408*x^6 + 7328*x^5 + 208512*x^4 - 219292*x^3 + 2113070*x^2 - 1950049*x + 6498881)
 

\( x^{12} - x^{11} + 37 x^{10} + 21 x^{9} + 997 x^{8} + 268 x^{7} + 16408 x^{6} + 7328 x^{5} + \cdots + 6498881 \) Copy content Toggle raw display

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

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(800667821812475251953125\) \(\medspace = 5^{9}\cdot 17^{6}\cdot 19^{8}\) 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:  \(98.16\)
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:  $5^{3/4}17^{1/2}19^{2/3}\approx 98.16447869097495$
Ramified primes:   \(5\), \(17\), \(19\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1615=5\cdot 17\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1615}(1,·)$, $\chi_{1615}(324,·)$, $\chi_{1615}(1223,·)$, $\chi_{1615}(1512,·)$, $\chi_{1615}(239,·)$, $\chi_{1615}(596,·)$, $\chi_{1615}(577,·)$, $\chi_{1615}(919,·)$, $\chi_{1615}(628,·)$, $\chi_{1615}(1531,·)$, $\chi_{1615}(1597,·)$, $\chi_{1615}(543,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.36125.1$^{2}$, 12.0.800667821812475251953125.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{29}a^{9}+\frac{12}{29}a^{8}-\frac{2}{29}a^{7}-\frac{5}{29}a^{6}-\frac{4}{29}a^{5}-\frac{9}{29}a^{4}+\frac{6}{29}a^{3}+\frac{4}{29}a^{2}+\frac{9}{29}a+\frac{14}{29}$, $\frac{1}{29}a^{10}-\frac{1}{29}a^{8}-\frac{10}{29}a^{7}-\frac{2}{29}a^{6}+\frac{10}{29}a^{5}-\frac{2}{29}a^{4}-\frac{10}{29}a^{3}-\frac{10}{29}a^{2}-\frac{7}{29}a+\frac{6}{29}$, $\frac{1}{23\!\cdots\!79}a^{11}+\frac{27\!\cdots\!32}{23\!\cdots\!79}a^{10}+\frac{51\!\cdots\!27}{23\!\cdots\!79}a^{9}-\frac{46\!\cdots\!66}{23\!\cdots\!79}a^{8}+\frac{74\!\cdots\!00}{23\!\cdots\!79}a^{7}+\frac{63\!\cdots\!56}{23\!\cdots\!79}a^{6}-\frac{44\!\cdots\!11}{23\!\cdots\!79}a^{5}+\frac{30\!\cdots\!63}{23\!\cdots\!79}a^{4}-\frac{57\!\cdots\!13}{23\!\cdots\!79}a^{3}+\frac{25\!\cdots\!28}{23\!\cdots\!79}a^{2}-\frac{79\!\cdots\!25}{23\!\cdots\!79}a+\frac{39\!\cdots\!02}{23\!\cdots\!79}$ 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

$C_{2834}$, which has order $2834$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Relative class number: $2834$ (assuming GRH)

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $5$
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:   $\frac{199820045643}{13\!\cdots\!09}a^{11}-\frac{65123645552}{47\!\cdots\!21}a^{10}-\frac{2034072231238}{13\!\cdots\!09}a^{9}-\frac{49857659840661}{13\!\cdots\!09}a^{8}-\frac{73530676700122}{13\!\cdots\!09}a^{7}-\frac{18\!\cdots\!26}{13\!\cdots\!09}a^{6}-\frac{30\!\cdots\!73}{13\!\cdots\!09}a^{5}-\frac{18\!\cdots\!56}{13\!\cdots\!09}a^{4}-\frac{12\!\cdots\!63}{13\!\cdots\!09}a^{3}-\frac{23\!\cdots\!85}{13\!\cdots\!09}a^{2}+\frac{14\!\cdots\!93}{13\!\cdots\!09}a-\frac{14\!\cdots\!94}{13\!\cdots\!09}$, $\frac{17\!\cdots\!95}{80\!\cdots\!51}a^{11}+\frac{49\!\cdots\!10}{80\!\cdots\!51}a^{10}+\frac{58\!\cdots\!70}{80\!\cdots\!51}a^{9}+\frac{23\!\cdots\!75}{80\!\cdots\!51}a^{8}+\frac{17\!\cdots\!80}{80\!\cdots\!51}a^{7}+\frac{54\!\cdots\!80}{80\!\cdots\!51}a^{6}+\frac{26\!\cdots\!29}{80\!\cdots\!51}a^{5}+\frac{77\!\cdots\!45}{80\!\cdots\!51}a^{4}+\frac{30\!\cdots\!25}{80\!\cdots\!51}a^{3}+\frac{34\!\cdots\!95}{80\!\cdots\!51}a^{2}+\frac{12\!\cdots\!40}{80\!\cdots\!51}a+\frac{38\!\cdots\!57}{80\!\cdots\!51}$, $\frac{22\!\cdots\!95}{80\!\cdots\!51}a^{11}+\frac{58\!\cdots\!66}{80\!\cdots\!51}a^{10}+\frac{78\!\cdots\!70}{80\!\cdots\!51}a^{9}+\frac{27\!\cdots\!30}{80\!\cdots\!51}a^{8}+\frac{23\!\cdots\!20}{80\!\cdots\!51}a^{7}+\frac{65\!\cdots\!50}{80\!\cdots\!51}a^{6}+\frac{35\!\cdots\!63}{80\!\cdots\!51}a^{5}+\frac{88\!\cdots\!00}{80\!\cdots\!51}a^{4}+\frac{42\!\cdots\!75}{80\!\cdots\!51}a^{3}+\frac{34\!\cdots\!70}{80\!\cdots\!51}a^{2}+\frac{17\!\cdots\!05}{80\!\cdots\!51}a+\frac{20\!\cdots\!46}{80\!\cdots\!51}$, $\frac{18\!\cdots\!87}{23\!\cdots\!79}a^{11}+\frac{65\!\cdots\!49}{23\!\cdots\!79}a^{10}+\frac{74\!\cdots\!34}{23\!\cdots\!79}a^{9}+\frac{27\!\cdots\!83}{23\!\cdots\!79}a^{8}+\frac{25\!\cdots\!00}{23\!\cdots\!79}a^{7}+\frac{64\!\cdots\!45}{23\!\cdots\!79}a^{6}+\frac{41\!\cdots\!96}{23\!\cdots\!79}a^{5}+\frac{84\!\cdots\!74}{23\!\cdots\!79}a^{4}+\frac{48\!\cdots\!09}{23\!\cdots\!79}a^{3}+\frac{39\!\cdots\!58}{23\!\cdots\!79}a^{2}+\frac{18\!\cdots\!73}{23\!\cdots\!79}a+\frac{53\!\cdots\!87}{23\!\cdots\!79}$, $\frac{33\!\cdots\!34}{23\!\cdots\!79}a^{11}-\frac{62\!\cdots\!41}{23\!\cdots\!79}a^{10}+\frac{28\!\cdots\!42}{80\!\cdots\!51}a^{9}+\frac{37\!\cdots\!30}{23\!\cdots\!79}a^{8}+\frac{24\!\cdots\!75}{23\!\cdots\!79}a^{7}-\frac{94\!\cdots\!36}{23\!\cdots\!79}a^{6}+\frac{27\!\cdots\!29}{23\!\cdots\!79}a^{5}+\frac{21\!\cdots\!56}{23\!\cdots\!79}a^{4}+\frac{50\!\cdots\!61}{23\!\cdots\!79}a^{3}-\frac{78\!\cdots\!52}{23\!\cdots\!79}a^{2}+\frac{31\!\cdots\!50}{23\!\cdots\!79}a-\frac{33\!\cdots\!99}{23\!\cdots\!79}$ 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:  \( 1234.55163261 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1234.55163261 \cdot 2834}{2\cdot\sqrt{800667821812475251953125}}\cr\approx \mathstrut & 0.120290721858 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 37*x^10 + 21*x^9 + 997*x^8 + 268*x^7 + 16408*x^6 + 7328*x^5 + 208512*x^4 - 219292*x^3 + 2113070*x^2 - 1950049*x + 6498881)
 
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^12 - x^11 + 37*x^10 + 21*x^9 + 997*x^8 + 268*x^7 + 16408*x^6 + 7328*x^5 + 208512*x^4 - 219292*x^3 + 2113070*x^2 - 1950049*x + 6498881, 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^12 - x^11 + 37*x^10 + 21*x^9 + 997*x^8 + 268*x^7 + 16408*x^6 + 7328*x^5 + 208512*x^4 - 219292*x^3 + 2113070*x^2 - 1950049*x + 6498881);
 
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^12 - x^11 + 37*x^10 + 21*x^9 + 997*x^8 + 268*x^7 + 16408*x^6 + 7328*x^5 + 208512*x^4 - 219292*x^3 + 2113070*x^2 - 1950049*x + 6498881);
 
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

$C_{12}$ (as 12T1):

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 cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.361.1, 4.0.36125.1, 6.6.16290125.1

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 ${\href{/padicField/2.12.0.1}{12} }$ ${\href{/padicField/3.12.0.1}{12} }$ R ${\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.12.0.1}{12} }$ R R ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(5\) Copy content Toggle raw display 5.12.9.1$x^{12} - 30 x^{8} + 225 x^{4} + 1125$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(17\) Copy content Toggle raw display 17.12.6.2$x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
\(19\) Copy content Toggle raw display 19.6.4.3$x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$