Properties

Label 20.20.386...584.1
Degree $20$
Signature $[20, 0]$
Discriminant $3.864\times 10^{36}$
Root discriminant $67.51$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^4:C_5$ (as 20T23)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 90*x^18 + 3201*x^16 - 57910*x^14 + 572604*x^12 - 3106616*x^10 + 8975251*x^8 - 13405134*x^6 + 10063119*x^4 - 3369434*x^2 + 326041)
 
gp: K = bnfinit(x^20 - 90*x^18 + 3201*x^16 - 57910*x^14 + 572604*x^12 - 3106616*x^10 + 8975251*x^8 - 13405134*x^6 + 10063119*x^4 - 3369434*x^2 + 326041, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![326041, 0, -3369434, 0, 10063119, 0, -13405134, 0, 8975251, 0, -3106616, 0, 572604, 0, -57910, 0, 3201, 0, -90, 0, 1]);
 

\( x^{20} - 90 x^{18} + 3201 x^{16} - 57910 x^{14} + 572604 x^{12} - 3106616 x^{10} + 8975251 x^{8} - 13405134 x^{6} + 10063119 x^{4} - 3369434 x^{2} + 326041 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[20, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3863718884402440860826881422560067584\)\(\medspace = 2^{30}\cdot 11^{16}\cdot 23^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $67.51$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 11, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4745312041700167413507567716} a^{18} - \frac{134926767356754744237838762}{1186328010425041853376891929} a^{16} - \frac{311619547720338764658489703}{2372656020850083706753783858} a^{14} + \frac{92538339562264785203735819}{2372656020850083706753783858} a^{12} - \frac{1}{4} a^{11} - \frac{601265767157120599929189281}{4745312041700167413507567716} a^{10} - \frac{1}{4} a^{9} - \frac{1064386708510399941219724129}{2372656020850083706753783858} a^{8} - \frac{112071576109266484473429466}{1186328010425041853376891929} a^{6} - \frac{1}{4} a^{5} + \frac{540388929091193477917593263}{2372656020850083706753783858} a^{4} + \frac{1}{4} a^{3} - \frac{2366406485251222549978269541}{4745312041700167413507567716} a^{2} + \frac{1}{4} a - \frac{1108726692070393574235080759}{4745312041700167413507567716}$, $\frac{1}{2709573175810795593112821165836} a^{19} + \frac{3424057263918370815892837025}{677393293952698898278205291459} a^{17} - \frac{198740016836422667043257931549}{2709573175810795593112821165836} a^{15} - \frac{1}{4} a^{14} - \frac{164714516769956288048980506493}{2709573175810795593112821165836} a^{13} - \frac{1}{4} a^{12} + \frac{115366348132863020404282368437}{1354786587905397796556410582918} a^{11} - \frac{1}{4} a^{10} - \frac{188504212355667012774768648911}{1354786587905397796556410582918} a^{9} - \frac{1}{2} a^{8} + \frac{990135602400472881631811042851}{2709573175810795593112821165836} a^{7} + \frac{1}{4} a^{6} - \frac{139123346760396821032826005026}{677393293952698898278205291459} a^{5} + \frac{181213165976324858392514120734}{677393293952698898278205291459} a^{3} + \frac{1}{4} a^{2} + \frac{657264518434650510910369034251}{1354786587905397796556410582918} a + \frac{1}{4}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 969444403068 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{20}\cdot(2\pi)^{0}\cdot 969444403068 \cdot 1}{2\sqrt{3863718884402440860826881422560067584}}\approx 0.258577102702887$ (assuming GRH)

Galois group

$C_2^4:C_5$ (as 20T23):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 80
The 8 conjugacy class representatives for $C_2^4:C_5$
Character table for $C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.116117348402176.2, 10.10.116117348402176.1, 10.10.61426077304751104.1

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 16 sibling: data not computed
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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$23$23.2.1.1$x^{2} - 23$$2$$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$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$