↑

Properties

Label 20.12.8227018811...0144.8
Degree $20$
Signature $[12, 4]$
Discriminant $2^{40}\cdot 11^{17}\cdot 23^{6}$
Root discriminant $78.66$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T326

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5819, 0, 93104, 0, 105754, 0, -1009976, 0, 1283953, 0, -506320, 0, -2503, 0, 23816, 0, 131, 0, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 60*x^18 + 131*x^16 + 23816*x^14 - 2503*x^12 - 506320*x^10 + 1283953*x^8 - 1009976*x^6 + 105754*x^4 + 93104*x^2 + 5819)
 
gp: K = bnfinit(x^20 - 60*x^18 + 131*x^16 + 23816*x^14 - 2503*x^12 - 506320*x^10 + 1283953*x^8 - 1009976*x^6 + 105754*x^4 + 93104*x^2 + 5819, 1)
 

Normalized defining polynomial

\( x^{20} - 60 x^{18} + 131 x^{16} + 23816 x^{14} - 2503 x^{12} - 506320 x^{10} + 1283953 x^{8} - 1009976 x^{6} + 105754 x^{4} + 93104 x^{2} + 5819 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(82270188117030423168911138645967110144=2^{40}\cdot 11^{17}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.66$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} + \frac{9}{23} a^{14} - \frac{7}{23} a^{12} + \frac{11}{23} a^{10} + \frac{4}{23} a^{8} + \frac{2}{23} a^{6} + \frac{1}{23} a^{4}$, $\frac{1}{23} a^{17} + \frac{9}{23} a^{15} - \frac{7}{23} a^{13} + \frac{11}{23} a^{11} + \frac{4}{23} a^{9} + \frac{2}{23} a^{7} + \frac{1}{23} a^{5}$, $\frac{1}{49950773191438835888187165346943} a^{18} - \frac{695558194191465736206572325558}{49950773191438835888187165346943} a^{16} - \frac{62686375973411581508965002365}{2171772747453862429921181102041} a^{14} - \frac{4888024054871931176233500753076}{49950773191438835888187165346943} a^{12} + \frac{24824388702341727337612840343247}{49950773191438835888187165346943} a^{10} - \frac{703089818278013171135102382426}{2171772747453862429921181102041} a^{8} + \frac{676896591172919389733010446584}{2171772747453862429921181102041} a^{6} + \frac{24133910408285164776455189068151}{49950773191438835888187165346943} a^{4} + \frac{766880148986553829631179259111}{2171772747453862429921181102041} a^{2} - \frac{1013018581839507879496326951967}{2171772747453862429921181102041}$, $\frac{1}{49950773191438835888187165346943} a^{19} - \frac{695558194191465736206572325558}{49950773191438835888187165346943} a^{17} - \frac{62686375973411581508965002365}{2171772747453862429921181102041} a^{15} - \frac{4888024054871931176233500753076}{49950773191438835888187165346943} a^{13} + \frac{24824388702341727337612840343247}{49950773191438835888187165346943} a^{11} - \frac{703089818278013171135102382426}{2171772747453862429921181102041} a^{9} + \frac{676896591172919389733010446584}{2171772747453862429921181102041} a^{7} + \frac{24133910408285164776455189068151}{49950773191438835888187165346943} a^{5} + \frac{766880148986553829631179259111}{2171772747453862429921181102041} a^{3} - \frac{1013018581839507879496326951967}{2171772747453862429921181102041} a$

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:  $15$
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:  \( 747432165015 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T326:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n326 are not computed
Character table for t20n326 is not computed

Intermediate fields

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

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$$\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}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$