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Properties

Label 11.3.102...000.1
Degree $11$
Signature $[3, 4]$
Discriminant $1.030\times 10^{27}$
Root discriminant \(285.56\)
Ramified primes $2,3,5,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{11}$ (as 11T8)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 55*x^9 + 1210*x^7 - 13310*x^5 + 73205*x^3 - 161051*x - 37500)
 
gp: K = bnfinit(y^11 - 55*y^9 + 1210*y^7 - 13310*y^5 + 73205*y^3 - 161051*y - 37500, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 55*x^9 + 1210*x^7 - 13310*x^5 + 73205*x^3 - 161051*x - 37500);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 - 55*x^9 + 1210*x^7 - 13310*x^5 + 73205*x^3 - 161051*x - 37500)
 

\( x^{11} - 55x^{9} + 1210x^{7} - 13310x^{5} + 73205x^{3} - 161051x - 37500 \) Copy content Toggle raw display

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

Invariants

Degree:  $11$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[3, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1029561428983324050000000000\) \(\medspace = 2^{10}\cdot 3^{8}\cdot 5^{11}\cdot 11^{12}\) 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:  \(285.56\)
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:  $2^{47/40}3^{4/5}5^{143/100}11^{64/55}\approx 884.5792590790442$
Ramified primes:   \(2\), \(3\), \(5\), \(11\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{5}a^{4}-\frac{1}{5}a^{2}$, $\frac{1}{25}a^{5}-\frac{2}{25}a^{3}+\frac{1}{25}a$, $\frac{1}{50}a^{6}-\frac{1}{50}a^{5}-\frac{1}{25}a^{4}+\frac{1}{25}a^{3}+\frac{1}{50}a^{2}-\frac{1}{50}a$, $\frac{1}{250}a^{7}-\frac{3}{250}a^{5}+\frac{3}{250}a^{3}-\frac{1}{250}a$, $\frac{1}{250}a^{8}+\frac{1}{125}a^{6}-\frac{1}{50}a^{5}-\frac{7}{250}a^{4}+\frac{1}{25}a^{3}+\frac{2}{125}a^{2}-\frac{1}{50}a$, $\frac{1}{1250}a^{9}+\frac{1}{1250}a^{7}-\frac{9}{1250}a^{5}+\frac{11}{1250}a^{3}-\frac{2}{625}a$, $\frac{1}{1250}a^{10}+\frac{1}{1250}a^{8}-\frac{9}{1250}a^{6}+\frac{11}{1250}a^{4}-\frac{2}{625}a^{2}$ 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

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:  $6$
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:   $\frac{2834389}{1250}a^{10}-\frac{67596}{125}a^{9}-\frac{155730821}{1250}a^{8}+\frac{7426603}{250}a^{7}+\frac{3420880679}{1250}a^{6}-\frac{163191267}{250}a^{5}-\frac{37534027701}{1250}a^{4}+\frac{1791441743}{250}a^{3}+\frac{102690761102}{625}a^{2}-\frac{9810014137}{250}a-355872931$, $\frac{2438128057}{1250}a^{10}+\frac{40428063}{5}a^{9}-\frac{46099716684}{625}a^{8}-\frac{76440705801}{250}a^{7}+\frac{682874490481}{625}a^{6}+\frac{1132315169073}{250}a^{5}-\frac{4490998085024}{625}a^{4}-\frac{7446793512393}{250}a^{3}+\frac{24133531598147}{1250}a^{2}+\frac{20008628304221}{250}a+17644586441$, $\frac{30515281}{1250}a^{10}+\frac{119480397}{1250}a^{9}-\frac{615501487}{625}a^{8}-\frac{2485692804}{625}a^{7}+\frac{8875484943}{625}a^{6}+\frac{36917973946}{625}a^{5}-\frac{56851888487}{625}a^{4}-\frac{241230438399}{625}a^{3}+\frac{289757634781}{1250}a^{2}+\frac{1261596421617}{1250}a+223035089$, $\frac{2099}{1250}a^{10}+\frac{15163}{1250}a^{9}-\frac{47893}{625}a^{8}-\frac{832447}{1250}a^{7}+\frac{660207}{625}a^{6}+\frac{17551013}{1250}a^{5}+\frac{401317}{625}a^{4}-\frac{169156587}{1250}a^{3}-\frac{153255611}{1250}a^{2}+\frac{313685804}{625}a+718393$, $\frac{1539939131}{1250}a^{10}+\frac{6384498209}{1250}a^{9}-\frac{29118359877}{625}a^{8}-\frac{120714483483}{625}a^{7}+\frac{862726378351}{1250}a^{6}+\frac{3576110146469}{1250}a^{5}-\frac{2837174187107}{625}a^{4}-\frac{11758512107813}{625}a^{3}+\frac{7623288628243}{625}a^{2}+\frac{31592730404582}{625}a+11143806769$, $\frac{324238}{625}a^{10}-\frac{435141}{250}a^{9}-\frac{28790339}{1250}a^{8}+\frac{1994174}{25}a^{7}+\frac{228095543}{625}a^{6}-\frac{323316859}{250}a^{5}-\frac{3170149559}{1250}a^{4}+\frac{1151517681}{125}a^{3}+\frac{3772666418}{625}a^{2}-\frac{2776169176}{125}a-5519533$ 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:  \( 16422422385.4 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 16422422385.4 \cdot 1}{2\cdot\sqrt{1029561428983324050000000000}}\cr\approx \mathstrut & 3.19073363734 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^11 - 55*x^9 + 1210*x^7 - 13310*x^5 + 73205*x^3 - 161051*x - 37500)
 
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^11 - 55*x^9 + 1210*x^7 - 13310*x^5 + 73205*x^3 - 161051*x - 37500, 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^11 - 55*x^9 + 1210*x^7 - 13310*x^5 + 73205*x^3 - 161051*x - 37500);
 
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^11 - 55*x^9 + 1210*x^7 - 13310*x^5 + 73205*x^3 - 161051*x - 37500);
 
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

$S_{11}$ (as 11T8):

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 non-solvable group of order 39916800
The 56 conjugacy class representatives for $S_{11}$
Character table for $S_{11}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 22 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 R ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ R ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.11.0.1}{11} }$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.10.10.16$x^{10} + 2 x^{2} + 2 x + 2$$10$$1$$10$$(C_2^4 : C_5):C_4$$[6/5, 6/5, 6/5, 6/5]_{5}^{4}$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.10.8.1$x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 598 x^{5} + 750 x^{4} + 640 x^{3} + 280 x^{2} + 40 x + 17$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.5.6.4$x^{5} + 20 x^{2} + 5$$5$$1$$6$$F_5$$[3/2]_{2}^{2}$
5.5.5.4$x^{5} + 10 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
\(11\) Copy content Toggle raw display 11.11.12.3$x^{11} + 77 x^{2} + 11$$11$$1$$12$$C_{11}:C_5$$[6/5]_{5}$