Properties

Label 20.0.78125000000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{23}$
Root discriminant $11.08$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois Group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 5, 15, -30, -21, 75, 15, -120, -5, 141, -5, -120, 15, 75, -21, -30, 15, 5, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 5*x^18 + 15*x^17 - 30*x^16 - 21*x^15 + 75*x^14 + 15*x^13 - 120*x^12 - 5*x^11 + 141*x^10 - 5*x^9 - 120*x^8 + 15*x^7 + 75*x^6 - 21*x^5 - 30*x^4 + 15*x^3 + 5*x^2 - 5*x + 1)
gp: K = bnfinit(x^20 - 5*x^19 + 5*x^18 + 15*x^17 - 30*x^16 - 21*x^15 + 75*x^14 + 15*x^13 - 120*x^12 - 5*x^11 + 141*x^10 - 5*x^9 - 120*x^8 + 15*x^7 + 75*x^6 - 21*x^5 - 30*x^4 + 15*x^3 + 5*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 5 x^{19} \) \(\mathstrut +\mathstrut 5 x^{18} \) \(\mathstrut +\mathstrut 15 x^{17} \) \(\mathstrut -\mathstrut 30 x^{16} \) \(\mathstrut -\mathstrut 21 x^{15} \) \(\mathstrut +\mathstrut 75 x^{14} \) \(\mathstrut +\mathstrut 15 x^{13} \) \(\mathstrut -\mathstrut 120 x^{12} \) \(\mathstrut -\mathstrut 5 x^{11} \) \(\mathstrut +\mathstrut 141 x^{10} \) \(\mathstrut -\mathstrut 5 x^{9} \) \(\mathstrut -\mathstrut 120 x^{8} \) \(\mathstrut +\mathstrut 15 x^{7} \) \(\mathstrut +\mathstrut 75 x^{6} \) \(\mathstrut -\mathstrut 21 x^{5} \) \(\mathstrut -\mathstrut 30 x^{4} \) \(\mathstrut +\mathstrut 15 x^{3} \) \(\mathstrut +\mathstrut 5 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 10]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(781250000000000000000=2^{16}\cdot 5^{23}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.08$
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 Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{431} a^{18} - \frac{86}{431} a^{17} + \frac{74}{431} a^{16} + \frac{141}{431} a^{15} + \frac{112}{431} a^{14} - \frac{183}{431} a^{13} + \frac{132}{431} a^{12} - \frac{150}{431} a^{11} - \frac{170}{431} a^{10} + \frac{123}{431} a^{9} - \frac{170}{431} a^{8} - \frac{150}{431} a^{7} + \frac{132}{431} a^{6} - \frac{183}{431} a^{5} + \frac{112}{431} a^{4} + \frac{141}{431} a^{3} + \frac{74}{431} a^{2} - \frac{86}{431} a + \frac{1}{431}$, $\frac{1}{431} a^{19} + \frac{5}{431} a^{17} + \frac{40}{431} a^{16} + \frac{170}{431} a^{15} - \frac{33}{431} a^{14} - \frac{90}{431} a^{13} - \frac{4}{431} a^{12} - \frac{140}{431} a^{11} + \frac{157}{431} a^{10} + \frac{64}{431} a^{9} - \frac{116}{431} a^{8} + \frac{162}{431} a^{7} - \frac{37}{431} a^{6} - \frac{110}{431} a^{5} - \frac{140}{431} a^{4} + \frac{132}{431} a^{3} - \frac{187}{431} a^{2} - \frac{68}{431} a + \frac{86}{431}$

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

Class group and class number

Trivial Abelian group, order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $9$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( \frac{288}{431} a^{19} + \frac{532}{431} a^{18} - \frac{6384}{431} a^{17} + \frac{5633}{431} a^{16} + \frac{23980}{431} a^{15} - \frac{30948}{431} a^{14} - \frac{54316}{431} a^{13} + \frac{77692}{431} a^{12} + \frac{93656}{431} a^{11} - \frac{121080}{431} a^{10} - \frac{133356}{431} a^{9} + \frac{124408}{431} a^{8} + \frac{140980}{431} a^{7} - \frac{80076}{431} a^{6} - \frac{92832}{431} a^{5} + \frac{43400}{431} a^{4} + \frac{38896}{431} a^{3} - \frac{17936}{431} a^{2} - \frac{8444}{431} a + \frac{4612}{431} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{2038}{431} a^{19} - \frac{7860}{431} a^{18} + \frac{429}{431} a^{17} + \frac{34320}{431} a^{16} - \frac{23497}{431} a^{15} - \frac{80833}{431} a^{14} + \frac{72296}{431} a^{13} + \frac{135699}{431} a^{12} - \frac{118739}{431} a^{11} - \frac{180325}{431} a^{10} + \frac{124351}{431} a^{9} + \frac{176159}{431} a^{8} - \frac{85113}{431} a^{7} - \frac{107834}{431} a^{6} + \frac{54810}{431} a^{5} + \frac{42022}{431} a^{4} - \frac{27240}{431} a^{3} - \frac{6788}{431} a^{2} + \frac{7246}{431} a - \frac{1113}{431} \),  \( \frac{2785}{431} a^{19} - \frac{12735}{431} a^{18} + \frac{8361}{431} a^{17} + \frac{45664}{431} a^{16} - \frac{63234}{431} a^{15} - \frac{88598}{431} a^{14} + \frac{170525}{431} a^{13} + \frac{124074}{431} a^{12} - \frac{282954}{431} a^{11} - \frac{153619}{431} a^{10} + \frac{331956}{431} a^{9} + \frac{154956}{431} a^{8} - \frac{272852}{431} a^{7} - \frac{103596}{431} a^{6} + \frac{166545}{431} a^{5} + \frac{32770}{431} a^{4} - \frac{70365}{431} a^{3} + \frac{2646}{431} a^{2} + \frac{16677}{431} a - \frac{4672}{431} \),  \( \frac{5469}{431} a^{19} - \frac{23941}{431} a^{18} + \frac{12299}{431} a^{17} + \frac{90098}{431} a^{16} - \frac{107345}{431} a^{15} - \frac{184497}{431} a^{14} + \frac{294453}{431} a^{13} + \frac{274532}{431} a^{12} - \frac{485883}{431} a^{11} - \frac{348561}{431} a^{10} + \frac{556745}{431} a^{9} + \frac{346158}{431} a^{8} - \frac{442737}{431} a^{7} - \frac{221437}{431} a^{6} + \frac{271704}{431} a^{5} + \frac{73356}{431} a^{4} - \frac{118631}{431} a^{3} - \frac{595}{431} a^{2} + \frac{28115}{431} a - \frac{7450}{431} \),  \( \frac{4675}{431} a^{19} - \frac{19961}{431} a^{18} + \frac{8263}{431} a^{17} + \frac{78311}{431} a^{16} - \frac{83699}{431} a^{15} - \frac{167240}{431} a^{14} + \frac{234508}{431} a^{13} + \frac{260008}{431} a^{12} - \frac{387287}{431} a^{11} - \frac{339108}{431} a^{10} + \frac{436462}{431} a^{9} + \frac{341788}{431} a^{8} - \frac{335673}{431} a^{7} - \frac{222258}{431} a^{6} + \frac{201348}{431} a^{5} + \frac{79889}{431} a^{4} - \frac{88949}{431} a^{3} - \frac{7992}{431} a^{2} + \frac{21270}{431} a - \frac{4518}{431} \),  \( \frac{3254}{431} a^{19} - \frac{16119}{431} a^{18} + \frac{14253}{431} a^{17} + \frac{54506}{431} a^{16} - \frac{96022}{431} a^{15} - \frac{93459}{431} a^{14} + \frac{255816}{431} a^{13} + \frac{107803}{431} a^{12} - \frac{431484}{431} a^{11} - \frac{115002}{431} a^{10} + \frac{526728}{431} a^{9} + \frac{114239}{431} a^{8} - \frac{460333}{431} a^{7} - \frac{78021}{431} a^{6} + \frac{289004}{431} a^{5} + \frac{11344}{431} a^{4} - \frac{124854}{431} a^{3} + \frac{16223}{431} a^{2} + \frac{29277}{431} a - \frac{9529}{431} \),  \( \frac{7450}{431} a^{19} - \frac{31781}{431} a^{18} + \frac{13309}{431} a^{17} + \frac{124049}{431} a^{16} - \frac{133402}{431} a^{15} - \frac{263795}{431} a^{14} + \frac{374253}{431} a^{13} + \frac{406203}{431} a^{12} - \frac{619468}{431} a^{11} - \frac{523133}{431} a^{10} + \frac{701889}{431} a^{9} + \frac{519495}{431} a^{8} - \frac{547842}{431} a^{7} - \frac{330987}{431} a^{6} + \frac{337313}{431} a^{5} + \frac{115254}{431} a^{4} - \frac{150144}{431} a^{3} - \frac{6881}{431} a^{2} + \frac{36655}{431} a - \frac{8704}{431} \),  \( \frac{7270}{431} a^{19} - \frac{30499}{431} a^{18} + \frac{11631}{431} a^{17} + \frac{119052}{431} a^{16} - \frac{122453}{431} a^{15} - \frac{253053}{431} a^{14} + \frac{342039}{431} a^{13} + \frac{389093}{431} a^{12} - \frac{559862}{431} a^{11} - \frac{496079}{431} a^{10} + \frac{625228}{431} a^{9} + \frac{479750}{431} a^{8} - \frac{482687}{431} a^{7} - \frac{291299}{431} a^{6} + \frac{304820}{431} a^{5} + \frac{99139}{431} a^{4} - \frac{135372}{431} a^{3} - \frac{3343}{431} a^{2} + \frac{32601}{431} a - \frac{8679}{431} \),  \( \frac{1319}{431} a^{19} - \frac{4403}{431} a^{18} - \frac{2647}{431} a^{17} + \frac{23466}{431} a^{16} - \frac{3952}{431} a^{15} - \frac{64287}{431} a^{14} + \frac{22868}{431} a^{13} + \frac{120368}{431} a^{12} - \frac{38824}{431} a^{11} - \frac{169317}{431} a^{10} + \frac{29446}{431} a^{9} + \frac{171402}{431} a^{8} - \frac{2095}{431} a^{7} - \frac{111075}{431} a^{6} - \frac{495}{431} a^{5} + \frac{49732}{431} a^{4} - \frac{3216}{431} a^{3} - \frac{13468}{431} a^{2} + \frac{1920}{431} a + \frac{850}{431} \),  \( \frac{1459}{431} a^{19} - \frac{7050}{431} a^{18} + \frac{5885}{431} a^{17} + \frac{23690}{431} a^{16} - \frac{39611}{431} a^{15} - \frac{41259}{431} a^{14} + \frac{104614}{431} a^{13} + \frac{48830}{431} a^{12} - \frac{173833}{431} a^{11} - \frac{52060}{431} a^{10} + \frac{208044}{431} a^{9} + \frac{47869}{431} a^{8} - \frac{177576}{431} a^{7} - \frac{27332}{431} a^{6} + \frac{111638}{431} a^{5} - \frac{836}{431} a^{4} - \frac{48936}{431} a^{3} + \frac{9282}{431} a^{2} + \frac{11438}{431} a - \frac{4411}{431} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 525.931286845 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.50000.1 x5, 10.2.12500000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.50000.1
Degree 10 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
5Data not computed