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Properties

Label 10.10.10368641602001.1
Degree $10$
Signature $[10, 0]$
Discriminant $1.037\times 10^{13}$
Root discriminant \(20.02\)
Ramified prime $401$
Class number $1$
Class group trivial
Galois group $D_5$ (as 10T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 20*x^8 + 2*x^7 + 69*x^6 - x^5 - 69*x^4 + 2*x^3 + 20*x^2 - 2*x - 1)
 
gp: K = bnfinit(y^10 - 2*y^9 - 20*y^8 + 2*y^7 + 69*y^6 - y^5 - 69*y^4 + 2*y^3 + 20*y^2 - 2*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 2*x^9 - 20*x^8 + 2*x^7 + 69*x^6 - x^5 - 69*x^4 + 2*x^3 + 20*x^2 - 2*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 2*x^9 - 20*x^8 + 2*x^7 + 69*x^6 - x^5 - 69*x^4 + 2*x^3 + 20*x^2 - 2*x - 1)
 

\( x^{10} - 2x^{9} - 20x^{8} + 2x^{7} + 69x^{6} - x^{5} - 69x^{4} + 2x^{3} + 20x^{2} - 2x - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[10, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(10368641602001\) \(\medspace = 401^{5}\) 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:  \(20.02\)
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:  $401^{1/2}\approx 20.024984394500787$
Ramified primes:   \(401\) 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:  \(\Q(\sqrt{401}) \)
$\card{ \Gal(K/\Q) }$:  $10$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{27}a^{8}+\frac{1}{9}a^{7}-\frac{4}{27}a^{6}+\frac{4}{9}a^{5}-\frac{1}{27}a^{4}+\frac{2}{9}a^{3}-\frac{4}{27}a^{2}-\frac{4}{9}a-\frac{8}{27}$, $\frac{1}{27}a^{9}-\frac{4}{27}a^{7}-\frac{1}{9}a^{6}-\frac{1}{27}a^{5}+\frac{1}{3}a^{4}-\frac{13}{27}a^{3}+\frac{10}{27}a-\frac{1}{9}$ 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:  $3$

Class group and class number

Trivial group, which has order $1$

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:  $9$
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:   $\frac{25}{27}a^{9}-\frac{32}{27}a^{8}-\frac{529}{27}a^{7}-\frac{316}{27}a^{6}+\frac{1607}{27}a^{5}+\frac{1076}{27}a^{4}-\frac{1309}{27}a^{3}-\frac{781}{27}a^{2}+\frac{220}{27}a+\frac{82}{27}$, $\frac{41}{27}a^{9}-\frac{70}{27}a^{8}-\frac{833}{27}a^{7}-\frac{176}{27}a^{6}+\frac{2629}{27}a^{5}+\frac{727}{27}a^{4}-\frac{2168}{27}a^{3}-\frac{566}{27}a^{2}+\frac{332}{27}a+\frac{50}{27}$, $\frac{7}{27}a^{9}-\frac{7}{27}a^{8}-\frac{148}{27}a^{7}-\frac{137}{27}a^{6}+\frac{377}{27}a^{5}+\frac{466}{27}a^{4}-\frac{124}{27}a^{3}-\frac{413}{27}a^{2}-\frac{80}{27}a+\frac{53}{27}$, $\frac{47}{27}a^{9}-\frac{77}{27}a^{8}-\frac{968}{27}a^{7}-\frac{256}{27}a^{6}+\frac{3151}{27}a^{5}+\frac{1103}{27}a^{4}-\frac{2810}{27}a^{3}-\frac{952}{27}a^{2}+\frac{521}{27}a+\frac{106}{27}$, $\frac{7}{27}a^{9}-\frac{8}{27}a^{8}-\frac{151}{27}a^{7}-\frac{106}{27}a^{6}+\frac{473}{27}a^{5}+\frac{359}{27}a^{4}-\frac{427}{27}a^{3}-\frac{220}{27}a^{2}+\frac{67}{27}a+\frac{7}{27}$, $2a^{9}-\frac{28}{9}a^{8}-\frac{124}{3}a^{7}-\frac{131}{9}a^{6}+\frac{392}{3}a^{5}+\frac{523}{9}a^{4}-\frac{326}{3}a^{3}-\frac{455}{9}a^{2}+\frac{49}{3}a+\frac{44}{9}$, $\frac{23}{27}a^{9}-\frac{5}{3}a^{8}-\frac{452}{27}a^{7}+\frac{1}{9}a^{6}+\frac{1399}{27}a^{5}+\frac{13}{3}a^{4}-\frac{983}{27}a^{3}-\frac{22}{3}a^{2}+\frac{32}{27}a+\frac{25}{9}$, $\frac{22}{27}a^{9}-\frac{50}{27}a^{8}-\frac{427}{27}a^{7}+\frac{161}{27}a^{6}+\frac{1484}{27}a^{5}-\frac{400}{27}a^{4}-\frac{1396}{27}a^{3}+\frac{335}{27}a^{2}+\frac{307}{27}a-\frac{98}{27}$, $\frac{4}{9}a^{9}-\frac{11}{27}a^{8}-\frac{29}{3}a^{7}-\frac{244}{27}a^{6}+28a^{5}+\frac{839}{27}a^{4}-\frac{197}{9}a^{3}-\frac{694}{27}a^{2}+\frac{17}{3}a+\frac{43}{27}$ Copy content Toggle raw display
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:  \( 1552.90547638 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 1552.90547638 \cdot 1}{2\cdot\sqrt{10368641602001}}\cr\approx \mathstrut & 0.246918739316 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 20*x^8 + 2*x^7 + 69*x^6 - x^5 - 69*x^4 + 2*x^3 + 20*x^2 - 2*x - 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^10 - 2*x^9 - 20*x^8 + 2*x^7 + 69*x^6 - x^5 - 69*x^4 + 2*x^3 + 20*x^2 - 2*x - 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^10 - 2*x^9 - 20*x^8 + 2*x^7 + 69*x^6 - x^5 - 69*x^4 + 2*x^3 + 20*x^2 - 2*x - 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^10 - 2*x^9 - 20*x^8 + 2*x^7 + 69*x^6 - x^5 - 69*x^4 + 2*x^3 + 20*x^2 - 2*x - 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

$D_5$ (as 10T2):

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 10
The 4 conjugacy class representatives for $D_5$
Character table for $D_5$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5

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 5 sibling: 5.5.160801.1
Minimal sibling: 5.5.160801.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.2.0.1}{2} }^{5}$ ${\href{/padicField/5.5.0.1}{5} }^{2}$ ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{5}$ ${\href{/padicField/17.2.0.1}{2} }^{5}$ ${\href{/padicField/19.2.0.1}{2} }^{5}$ ${\href{/padicField/23.2.0.1}{2} }^{5}$ ${\href{/padicField/29.5.0.1}{5} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{5}$ ${\href{/padicField/37.2.0.1}{2} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{5}$ ${\href{/padicField/59.2.0.1}{2} }^{5}$

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
\(401\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.401.2t1.a.a$1$ $ 401 $ \(\Q(\sqrt{401}) \) $C_2$ (as 2T1) $1$ $1$
*2 2.401.5t2.b.a$2$ $ 401 $ 10.10.10368641602001.1 $D_5$ (as 10T2) $1$ $2$
*2 2.401.5t2.b.b$2$ $ 401 $ 10.10.10368641602001.1 $D_5$ (as 10T2) $1$ $2$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.