Properties

Label 10.0.224415603.1
Degree $10$
Signature $[0, 5]$
Discriminant $-224415603$
Root discriminant $6.84$
Ramified primes $3, 31$
Class number $1$
Class group trivial
Galois group $D_5\times C_5$ (as 10T6)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 2*x^8 - 3*x^7 + 3*x^6 - 7*x^5 + 8*x^4 - 7*x^3 + 7*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^10 + 2*x^8 - 3*x^7 + 3*x^6 - 7*x^5 + 8*x^4 - 7*x^3 + 7*x^2 - 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 7, -7, 8, -7, 3, -3, 2, 0, 1]);
 

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

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

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-224415603\)\(\medspace = -\,3^{5}\cdot 31^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $6.84$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 31$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $5$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( 2 a^{9} + a^{8} + 3 a^{7} - 4 a^{6} + 3 a^{5} - 8 a^{4} + 8 a^{3} - 4 a^{2} + 4 a - 1 \) (order $6$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{9} + 3 a^{7} - 2 a^{6} + 4 a^{5} - 9 a^{4} + 7 a^{3} - 9 a^{2} + 8 a - 2 \),  \( 2 a^{9} + 2 a^{8} + 4 a^{7} - 3 a^{6} + a^{5} - 9 a^{4} + 5 a^{3} - 3 a^{2} + 5 a - 2 \),  \( a^{7} + a^{5} - 2 a^{4} + 3 a^{3} - 3 a^{2} + 4 a - 2 \),  \( 2 a^{9} + 2 a^{8} + 5 a^{7} - 3 a^{6} + 2 a^{5} - 11 a^{4} + 8 a^{3} - 6 a^{2} + 10 a - 4 \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1.78149530531 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{5}\cdot 1.78149530531 \cdot 1}{6\sqrt{224415603}}\approx 0.194091380560$

Galois group

$C_5\times D_5$ (as 10T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 20 conjugacy class representatives for $D_5\times C_5$
Character table for $D_5\times C_5$

Intermediate fields

\(\Q(\sqrt{-3}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 sibling: data not computed
Degree 25 sibling: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }$ R ${\href{/padicField/5.2.0.1}{2} }^{5}$ ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }$ ${\href{/padicField/29.10.0.1}{10} }$ R ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ ${\href{/padicField/47.10.0.1}{10} }$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.10.0.1}{10} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$31$31.5.4.5$x^{5} - 74431$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.93.10t1.a.d$1$ $ 3 \cdot 31 $ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
1.31.5t1.a.d$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
1.93.10t1.a.b$1$ $ 3 \cdot 31 $ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
1.31.5t1.a.b$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
1.93.10t1.a.c$1$ $ 3 \cdot 31 $ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
1.93.10t1.a.a$1$ $ 3 \cdot 31 $ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
1.31.5t1.a.a$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
1.31.5t1.a.c$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
2.2883.5t2.a.b$2$ $ 3 \cdot 31^{2}$ 5.1.8311689.1 $D_{5}$ (as 5T2) $1$ $0$
2.2883.10t6.b.a$2$ $ 3 \cdot 31^{2}$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.2883.10t6.b.b$2$ $ 3 \cdot 31^{2}$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.93.10t6.b.b$2$ $ 3 \cdot 31 $ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.93.10t6.b.a$2$ $ 3 \cdot 31 $ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.2883.10t6.b.c$2$ $ 3 \cdot 31^{2}$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.93.10t6.b.c$2$ $ 3 \cdot 31 $ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.2883.5t2.a.a$2$ $ 3 \cdot 31^{2}$ 5.1.8311689.1 $D_{5}$ (as 5T2) $1$ $0$
2.2883.10t6.b.d$2$ $ 3 \cdot 31^{2}$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.93.10t6.b.d$2$ $ 3 \cdot 31 $ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$

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 *.