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Properties

Label 20.0.42375791100...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{23}\cdot 7^{8}\cdot 97^{2}$
Root discriminant $38.14$
Ramified primes $2, 5, 7, 97$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group 20T135

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3581, -5135, 24440, -14740, 24160, -34527, 13900, -3890, 9275, -1950, -3126, -1370, 2990, -360, -610, 122, 75, -5, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 - 5*x^17 + 75*x^16 + 122*x^15 - 610*x^14 - 360*x^13 + 2990*x^12 - 1370*x^11 - 3126*x^10 - 1950*x^9 + 9275*x^8 - 3890*x^7 + 13900*x^6 - 34527*x^5 + 24160*x^4 - 14740*x^3 + 24440*x^2 - 5135*x + 3581)
 
gp: K = bnfinit(x^20 - 10*x^18 - 5*x^17 + 75*x^16 + 122*x^15 - 610*x^14 - 360*x^13 + 2990*x^12 - 1370*x^11 - 3126*x^10 - 1950*x^9 + 9275*x^8 - 3890*x^7 + 13900*x^6 - 34527*x^5 + 24160*x^4 - 14740*x^3 + 24440*x^2 - 5135*x + 3581, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} - 5 x^{17} + 75 x^{16} + 122 x^{15} - 610 x^{14} - 360 x^{13} + 2990 x^{12} - 1370 x^{11} - 3126 x^{10} - 1950 x^{9} + 9275 x^{8} - 3890 x^{7} + 13900 x^{6} - 34527 x^{5} + 24160 x^{4} - 14740 x^{3} + 24440 x^{2} - 5135 x + 3581 \)

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:  \(42375791100781250000000000000000=2^{16}\cdot 5^{23}\cdot 7^{8}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.14$
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:  $2, 5, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{14} + \frac{2}{7} a^{13} - \frac{1}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{17} + \frac{2}{7} a^{15} + \frac{2}{7} a^{14} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{18} + \frac{2}{7} a^{15} + \frac{3}{7} a^{14} + \frac{2}{7} a^{13} - \frac{3}{7} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{3818511267592320163508096645226331447597643} a^{19} - \frac{215211275519879905750696155690593983164611}{3818511267592320163508096645226331447597643} a^{18} + \frac{69456417115603766354430748516899554979244}{3818511267592320163508096645226331447597643} a^{17} - \frac{42371672089104516964799203255406843020797}{3818511267592320163508096645226331447597643} a^{16} - \frac{225648324867484057955555312373952764307725}{3818511267592320163508096645226331447597643} a^{15} - \frac{1032575829296108276002786178468968123995252}{3818511267592320163508096645226331447597643} a^{14} + \frac{145714537628896550672647421415286345438388}{3818511267592320163508096645226331447597643} a^{13} + \frac{848806557985950232253538762065421630453749}{3818511267592320163508096645226331447597643} a^{12} + \frac{675992804302447620355377220050754590729995}{3818511267592320163508096645226331447597643} a^{11} - \frac{1906542859068664609448086911812452379548098}{3818511267592320163508096645226331447597643} a^{10} + \frac{84765052309929912926248403460179773880236}{545501609656045737644013806460904492513949} a^{9} - \frac{1650852852787119662070377845451693650806978}{3818511267592320163508096645226331447597643} a^{8} + \frac{269937257524353210121444489167140129539615}{545501609656045737644013806460904492513949} a^{7} + \frac{502355927647536110036027584814636001361344}{3818511267592320163508096645226331447597643} a^{6} - \frac{1285714747128708484348012863441421416234749}{3818511267592320163508096645226331447597643} a^{5} - \frac{651280285531882178463884139504498385762656}{3818511267592320163508096645226331447597643} a^{4} + \frac{1772895184216701262114449128690607131051495}{3818511267592320163508096645226331447597643} a^{3} - \frac{437351820570331409556860054844434074954951}{3818511267592320163508096645226331447597643} a^{2} - \frac{1881175229983333510709847757129400511721744}{3818511267592320163508096645226331447597643} a - \frac{764398086696702031655813477162527534271598}{3818511267592320163508096645226331447597643}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( 1633832.53547 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T135:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n135
Character table for t20n135 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.2450000.1, 10.10.30012500000000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$