↑

Properties

Label 20.8.44145664201...6624.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 401^{11}$
Root discriminant $38.22$
Ramified primes $2, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![288, -2688, 976, 10096, -2440, -9152, -8384, 2828, 12814, -2816, 1309, -4375, 713, 131, 433, -23, -125, 38, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 38*x^17 - 125*x^16 - 23*x^15 + 433*x^14 + 131*x^13 + 713*x^12 - 4375*x^11 + 1309*x^10 - 2816*x^9 + 12814*x^8 + 2828*x^7 - 8384*x^6 - 9152*x^5 - 2440*x^4 + 10096*x^3 + 976*x^2 - 2688*x + 288)
 
gp: K = bnfinit(x^20 - 3*x^19 + 38*x^17 - 125*x^16 - 23*x^15 + 433*x^14 + 131*x^13 + 713*x^12 - 4375*x^11 + 1309*x^10 - 2816*x^9 + 12814*x^8 + 2828*x^7 - 8384*x^6 - 9152*x^5 - 2440*x^4 + 10096*x^3 + 976*x^2 - 2688*x + 288, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 38 x^{17} - 125 x^{16} - 23 x^{15} + 433 x^{14} + 131 x^{13} + 713 x^{12} - 4375 x^{11} + 1309 x^{10} - 2816 x^{9} + 12814 x^{8} + 2828 x^{7} - 8384 x^{6} - 9152 x^{5} - 2440 x^{4} + 10096 x^{3} + 976 x^{2} - 2688 x + 288 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44145664201696349530694455706624=2^{10}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.22$
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:  $2, 401$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{3}{8} a^{10} - \frac{1}{8} a^{9} - \frac{3}{8} a^{8} - \frac{3}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{17} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{3}{16} a^{12} - \frac{1}{16} a^{11} - \frac{3}{16} a^{10} + \frac{5}{16} a^{9} - \frac{3}{16} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4187874892292427105741461237788298418672} a^{19} + \frac{688675466019134733512531515806488893}{73471489338463633434060723469970147696} a^{18} - \frac{33982677780593622085427079022175963}{4591968083653977089628795216873134231} a^{17} - \frac{2807092773546890346715752944016215377}{55103617003847725075545542602477610772} a^{16} + \frac{349485715656744009980612978895501091561}{4187874892292427105741461237788298418672} a^{15} - \frac{376590446740717260077089062345959799551}{4187874892292427105741461237788298418672} a^{14} + \frac{670521823462003444070406877316162600293}{4187874892292427105741461237788298418672} a^{13} + \frac{475031051478865341744960168802498135613}{4187874892292427105741461237788298418672} a^{12} - \frac{829008730042096504461392503198668215929}{4187874892292427105741461237788298418672} a^{11} + \frac{204346164513467092574326097142489192143}{4187874892292427105741461237788298418672} a^{10} + \frac{1271280899610354182700969587684730920839}{4187874892292427105741461237788298418672} a^{9} - \frac{31224963448561821166163743318676725123}{110207234007695450151091085204955221544} a^{8} - \frac{398871898453235378401263095137872766535}{1046968723073106776435365309447074604668} a^{7} + \frac{247932595544159120816124737419713774265}{2093937446146213552870730618894149209336} a^{6} - \frac{74115957576274759525534233431682370183}{523484361536553388217682654723537302334} a^{5} - \frac{11231720513046707110555766138133086659}{61586395474888633907962665261592623804} a^{4} + \frac{4558207356122599565900449412534950867}{27551808501923862537772771301238805386} a^{3} + \frac{49520892140501089769614085931561519556}{261742180768276694108841327361768651167} a^{2} + \frac{16036550993746082786179610527978704061}{261742180768276694108841327361768651167} a - \frac{36536879602765839759038384564916730940}{87247393589425564702947109120589550389}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 178050557.612 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n350 are not computed
Character table for t20n350 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
401Data not computed