Properties

Label 5.1.100000000.1
Degree $5$
Signature $[1, 2]$
Discriminant $2^{8}\cdot 5^{8}$
Root discriminant $39.81$
Ramified primes $2, 5$
Class number $3$
Class group $[3]$
Galois Group $A_5$ (as 5T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![216, 10, -20, -10, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 10*x^3 - 20*x^2 + 10*x + 216)
gp: K = bnfinit(x^5 - 10*x^3 - 20*x^2 + 10*x + 216, 1)

Normalized defining polynomial

\(x^{5} \) \(\mathstrut -\mathstrut 10 x^{3} \) \(\mathstrut -\mathstrut 20 x^{2} \) \(\mathstrut +\mathstrut 10 x \) \(\mathstrut +\mathstrut 216 \)

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

Invariants

Degree:  $5$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 2]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(100000000=2^{8}\cdot 5^{8}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $39.81$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{94} a^{4} + \frac{21}{47} a^{3} - \frac{16}{47} a^{2} + \frac{23}{47} a - \frac{16}{47}$

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

Class group and class number

$C_{3}$, which has order $3$

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

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $2$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{284}{47} a^{4} - \frac{480}{47} a^{3} - \frac{3636}{47} a^{2} + \frac{5074}{47} a + \frac{9289}{47} \),  \( \frac{207}{47} a^{4} + \frac{187}{47} a^{3} - \frac{4133}{47} a^{2} - \frac{1429}{47} a + \frac{21811}{47} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 254.736382083 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$A_5$ (as 5T4):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
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 6 sibling: 6.2.100000000.1
Degree 10 sibling: 10.2.10000000000000000.2
Degree 12 sibling: 12.0.2560000000000000000.23
Degree 15 sibling: Deg 15
Degree 20 sibling: Deg 20
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 R ${\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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];
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]])

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
$5$5.5.8.6$x^{5} + 5 x^{4} + 5$$5$$1$$8$$D_{5}$$[2]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
3.2e8_5e4.12t33.1c1$3$ $ 2^{8} \cdot 5^{4}$ $x^{5} - 10 x^{3} - 20 x^{2} + 10 x + 216$ $A_5$ (as 5T4) $1$ $-1$
3.2e8_5e4.12t33.1c2$3$ $ 2^{8} \cdot 5^{4}$ $x^{5} - 10 x^{3} - 20 x^{2} + 10 x + 216$ $A_5$ (as 5T4) $1$ $-1$
* 4.2e8_5e8.5t4.1c1$4$ $ 2^{8} \cdot 5^{8}$ $x^{5} - 10 x^{3} - 20 x^{2} + 10 x + 216$ $A_5$ (as 5T4) $1$ $0$
5.2e8_5e8.6t12.1c1$5$ $ 2^{8} \cdot 5^{8}$ $x^{5} - 10 x^{3} - 20 x^{2} + 10 x + 216$ $A_5$ (as 5T4) $1$ $1$

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 *.