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Properties

Label 20.0.14925473637...9232.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 23^{10}$
Root discriminant $32.26$
Ramified primes $2, 23$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
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![18496, -97920, 234400, -340128, 410364, -409424, 339544, -212512, 105208, -49080, 29132, -17560, 9133, -4136, 2080, -896, 286, -80, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 28*x^18 - 80*x^17 + 286*x^16 - 896*x^15 + 2080*x^14 - 4136*x^13 + 9133*x^12 - 17560*x^11 + 29132*x^10 - 49080*x^9 + 105208*x^8 - 212512*x^7 + 339544*x^6 - 409424*x^5 + 410364*x^4 - 340128*x^3 + 234400*x^2 - 97920*x + 18496)
 
gp: K = bnfinit(x^20 - 8*x^19 + 28*x^18 - 80*x^17 + 286*x^16 - 896*x^15 + 2080*x^14 - 4136*x^13 + 9133*x^12 - 17560*x^11 + 29132*x^10 - 49080*x^9 + 105208*x^8 - 212512*x^7 + 339544*x^6 - 409424*x^5 + 410364*x^4 - 340128*x^3 + 234400*x^2 - 97920*x + 18496, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 28 x^{18} - 80 x^{17} + 286 x^{16} - 896 x^{15} + 2080 x^{14} - 4136 x^{13} + 9133 x^{12} - 17560 x^{11} + 29132 x^{10} - 49080 x^{9} + 105208 x^{8} - 212512 x^{7} + 339544 x^{6} - 409424 x^{5} + 410364 x^{4} - 340128 x^{3} + 234400 x^{2} - 97920 x + 18496 \)

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:  \(1492547363720394483260280799232=2^{55}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.26$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 23$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{552} a^{14} + \frac{13}{276} a^{13} + \frac{61}{552} a^{12} - \frac{5}{138} a^{11} + \frac{31}{552} a^{10} + \frac{13}{276} a^{9} + \frac{73}{552} a^{8} + \frac{3}{46} a^{7} - \frac{13}{138} a^{6} + \frac{5}{138} a^{5} + \frac{31}{92} a^{4} - \frac{5}{23} a^{3} + \frac{65}{138} a^{2} + \frac{8}{23} a + \frac{2}{69}$, $\frac{1}{552} a^{15} + \frac{1}{92} a^{13} - \frac{19}{552} a^{12} - \frac{1}{552} a^{11} + \frac{2}{23} a^{10} + \frac{3}{92} a^{9} + \frac{1}{552} a^{8} + \frac{29}{138} a^{7} - \frac{1}{69} a^{6} + \frac{10}{69} a^{5} - \frac{21}{92} a^{4} + \frac{17}{138} a^{3} + \frac{7}{69} a^{2} - \frac{1}{69} a + \frac{17}{69}$, $\frac{1}{1104} a^{16} - \frac{1}{1104} a^{14} + \frac{1}{184} a^{13} - \frac{83}{1104} a^{12} - \frac{11}{138} a^{11} + \frac{77}{1104} a^{10} + \frac{13}{552} a^{9} - \frac{25}{552} a^{8} - \frac{65}{276} a^{7} - \frac{9}{92} a^{6} + \frac{37}{276} a^{5} + \frac{1}{138} a^{4} + \frac{43}{138} a^{3} + \frac{95}{276} a^{2} + \frac{28}{69} a - \frac{7}{69}$, $\frac{1}{9440304} a^{17} - \frac{995}{4720152} a^{16} - \frac{1977}{3146768} a^{15} - \frac{59}{1180038} a^{14} + \frac{65185}{3146768} a^{13} + \frac{356783}{4720152} a^{12} - \frac{286755}{3146768} a^{11} - \frac{263}{69414} a^{10} - \frac{195917}{2360076} a^{9} + \frac{247283}{2360076} a^{8} + \frac{573391}{2360076} a^{7} - \frac{164351}{2360076} a^{6} + \frac{347777}{2360076} a^{5} - \frac{42478}{196673} a^{4} + \frac{41753}{2360076} a^{3} - \frac{22521}{393346} a^{2} - \frac{102733}{590019} a - \frac{2241}{11569}$, $\frac{1}{18880608} a^{18} - \frac{1}{18880608} a^{17} + \frac{3623}{18880608} a^{16} + \frac{3149}{18880608} a^{15} + \frac{9235}{18880608} a^{14} + \frac{465145}{18880608} a^{13} - \frac{1248715}{18880608} a^{12} - \frac{133589}{1110624} a^{11} - \frac{1988}{590019} a^{10} + \frac{133303}{1573384} a^{9} - \frac{145135}{786692} a^{8} - \frac{282085}{1573384} a^{7} + \frac{1170713}{4720152} a^{6} + \frac{195305}{4720152} a^{5} + \frac{621543}{1573384} a^{4} - \frac{81001}{205224} a^{3} + \frac{30673}{393346} a^{2} + \frac{26905}{69414} a + \frac{11170}{34707}$, $\frac{1}{321989686779099064440600672} a^{19} - \frac{604158140003602101}{107329895593033021480200224} a^{18} - \frac{14817157924280611739}{321989686779099064440600672} a^{17} + \frac{82948070133126858744275}{321989686779099064440600672} a^{16} - \frac{229022380176483332699023}{321989686779099064440600672} a^{15} + \frac{195569087881206459426427}{321989686779099064440600672} a^{14} + \frac{4641001133112999753562145}{107329895593033021480200224} a^{13} - \frac{6408578010377428993635149}{107329895593033021480200224} a^{12} + \frac{3901466723385680371938161}{160994843389549532220300336} a^{11} - \frac{3301164512759663407716475}{40248710847387383055075084} a^{10} + \frac{3368238946925291254725839}{80497421694774766110150168} a^{9} - \frac{62487636684711160083365}{394595204386150814265442} a^{8} - \frac{11623569450015978473394701}{80497421694774766110150168} a^{7} + \frac{2060914424504235382632253}{26832473898258255370050056} a^{6} + \frac{15740034509438664552334393}{80497421694774766110150168} a^{5} - \frac{1209507084104114332455793}{26832473898258255370050056} a^{4} + \frac{2711910832172175667331033}{40248710847387383055075084} a^{3} - \frac{986977928890727911314398}{10062177711846845763768771} a^{2} - \frac{747016062722261762769066}{3354059237282281921256257} a - \frac{12704645205788017456696}{591892806579226221398163}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 38650073.2246 \) (assuming GRH)
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

\(\Q(\sqrt{2}) \), 4.0.1083392.5, 5.1.1083392.1 x5, 10.2.9389905805312.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.1083392.1
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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$