Properties

Label 10.2.18825957754321.1
Degree $10$
Signature $[2, 4]$
Discriminant $1.883\times 10^{13}$
Root discriminant $21.26$
Ramified prime $2083$
Class number $2$
Class group $[2]$
Galois group $A_{5}$ (as 10T7)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 12*x^8 - 23*x^7 + 33*x^6 - 32*x^5 + 20*x^4 - 8*x^3 + 2*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^10 - 4*x^9 + 12*x^8 - 23*x^7 + 33*x^6 - 32*x^5 + 20*x^4 - 8*x^3 + 2*x^2 - 3*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 2, -8, 20, -32, 33, -23, 12, -4, 1]);
 

\(x^{10} - 4 x^{9} + 12 x^{8} - 23 x^{7} + 33 x^{6} - 32 x^{5} + 20 x^{4} - 8 x^{3} + 2 x^{2} - 3 x + 1\)  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:  $[2, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(18825957754321\)\(\medspace = 2083^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $21.26$
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:  $2083$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
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

$C_{2}$, which has order $2$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{9} - 4 a^{8} + 12 a^{7} - 23 a^{6} + 33 a^{5} - 32 a^{4} + 20 a^{3} - 8 a^{2} + 2 a - 3 \),  \( a^{2} - a + 1 \),  \( a - 1 \),  \( a^{7} - 3 a^{6} + 8 a^{5} - 12 a^{4} + 13 a^{3} - 7 a^{2} \),  \( a^{7} - 3 a^{6} + 8 a^{5} - 13 a^{4} + 15 a^{3} - 11 a^{2} + 5 a - 1 \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 160.73086375 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{4}\cdot 160.73086375 \cdot 2}{2\sqrt{18825957754321}}\approx 0.23094055407$

Galois group

$A_5$ (as 10T7):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 60
The 5 conjugacy class representatives for $A_{5}$
Character table for $A_{5}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 5 sibling: 5.1.4338889.1
Degree 6 sibling: 6.2.4338889.1
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 sibling: data not computed
Degree 30 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 ${\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{2}$ ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.5.0.1}{5} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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
2083Data not computed