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Properties

Label 20.0.13806836326...5424.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{4}$
Root discriminant $36.06$
Ramified primes $2, 11, 23$
Class number $32$ (GRH)
Class group $[4, 8]$ (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, -414, 12646, -49020, 72747, -59590, 53696, -33322, 28503, -9186, 5848, 430, 1872, 740, 324, -52, 81, -18, 18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 18*x^18 - 18*x^17 + 81*x^16 - 52*x^15 + 324*x^14 + 740*x^13 + 1872*x^12 + 430*x^11 + 5848*x^10 - 9186*x^9 + 28503*x^8 - 33322*x^7 + 53696*x^6 - 59590*x^5 + 72747*x^4 - 49020*x^3 + 12646*x^2 - 414*x + 529)
 
gp: K = bnfinit(x^20 - 4*x^19 + 18*x^18 - 18*x^17 + 81*x^16 - 52*x^15 + 324*x^14 + 740*x^13 + 1872*x^12 + 430*x^11 + 5848*x^10 - 9186*x^9 + 28503*x^8 - 33322*x^7 + 53696*x^6 - 59590*x^5 + 72747*x^4 - 49020*x^3 + 12646*x^2 - 414*x + 529, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 18 x^{18} - 18 x^{17} + 81 x^{16} - 52 x^{15} + 324 x^{14} + 740 x^{13} + 1872 x^{12} + 430 x^{11} + 5848 x^{10} - 9186 x^{9} + 28503 x^{8} - 33322 x^{7} + 53696 x^{6} - 59590 x^{5} + 72747 x^{4} - 49020 x^{3} + 12646 x^{2} - 414 x + 529 \)

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:  \(13806836326351181066487331815424=2^{30}\cdot 11^{16}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.06$
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:  $2, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{67} a^{18} + \frac{5}{67} a^{17} - \frac{33}{67} a^{16} + \frac{9}{67} a^{15} - \frac{20}{67} a^{14} - \frac{24}{67} a^{13} + \frac{18}{67} a^{12} - \frac{10}{67} a^{11} - \frac{13}{67} a^{10} - \frac{6}{67} a^{8} + \frac{6}{67} a^{7} - \frac{12}{67} a^{6} + \frac{30}{67} a^{5} - \frac{23}{67} a^{4} - \frac{32}{67} a^{3} + \frac{29}{67} a^{2} + \frac{7}{67} a + \frac{9}{67}$, $\frac{1}{15104389194804078602250609636208231445847434523887} a^{19} - \frac{77848437831746118902902272331405771742588929179}{15104389194804078602250609636208231445847434523887} a^{18} - \frac{6054568474406341445588086314140921323128415107308}{15104389194804078602250609636208231445847434523887} a^{17} + \frac{2578021195673031525564013877545501793086836610200}{15104389194804078602250609636208231445847434523887} a^{16} + \frac{223209308459042967654176433540507964343209235025}{15104389194804078602250609636208231445847434523887} a^{15} + \frac{2913978859761733741903343380467335240644727131729}{15104389194804078602250609636208231445847434523887} a^{14} + \frac{628892915331053775906351610277891856376136148352}{15104389194804078602250609636208231445847434523887} a^{13} + \frac{2567344203316823430458791358113897788683832867139}{15104389194804078602250609636208231445847434523887} a^{12} + \frac{4406293023464195202897725189363672358415297140135}{15104389194804078602250609636208231445847434523887} a^{11} + \frac{4582317310363884059870136030375887264972076842260}{15104389194804078602250609636208231445847434523887} a^{10} - \frac{6424698267664653889652161827823922789406303527646}{15104389194804078602250609636208231445847434523887} a^{9} - \frac{5429102838242721469045346529218561111274559533438}{15104389194804078602250609636208231445847434523887} a^{8} + \frac{5917198547811703853534454661116278821150560142144}{15104389194804078602250609636208231445847434523887} a^{7} + \frac{41365069883384696961606639810284580631093772976}{15104389194804078602250609636208231445847434523887} a^{6} + \frac{6590760760346137593418775998460404202798718155332}{15104389194804078602250609636208231445847434523887} a^{5} - \frac{1082240733706114671191698100999889104814159695459}{15104389194804078602250609636208231445847434523887} a^{4} + \frac{2002202970515175382322564272778963096432740080182}{15104389194804078602250609636208231445847434523887} a^{3} + \frac{7092178330867676112662523114674979451591838205687}{15104389194804078602250609636208231445847434523887} a^{2} - \frac{5043628997611079449466976713139097292580445312408}{15104389194804078602250609636208231445847434523887} a - \frac{20067699664821667553047371611685266339140227206}{656712573687133852271765636356879628080323240169}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( 447363.730538 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.5048580365312.1, 10.0.116117348402176.1, 10.10.5048580365312.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$