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Properties

Label 20.0.64728923771...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{22}\cdot 23^{10}$
Root discriminant $49.04$
Ramified primes $2, 5, 23$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10740479, -1174080, 22642685, -1889530, 26512155, -246742, 17624360, 202090, 6965605, 61230, 1727545, 5430, 276395, 90, 28460, -4, 1815, 0, 65, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 65*x^18 + 1815*x^16 - 4*x^15 + 28460*x^14 + 90*x^13 + 276395*x^12 + 5430*x^11 + 1727545*x^10 + 61230*x^9 + 6965605*x^8 + 202090*x^7 + 17624360*x^6 - 246742*x^5 + 26512155*x^4 - 1889530*x^3 + 22642685*x^2 - 1174080*x + 10740479)
 
gp: K = bnfinit(x^20 + 65*x^18 + 1815*x^16 - 4*x^15 + 28460*x^14 + 90*x^13 + 276395*x^12 + 5430*x^11 + 1727545*x^10 + 61230*x^9 + 6965605*x^8 + 202090*x^7 + 17624360*x^6 - 246742*x^5 + 26512155*x^4 - 1889530*x^3 + 22642685*x^2 - 1174080*x + 10740479, 1)
 

Normalized defining polynomial

\( x^{20} + 65 x^{18} + 1815 x^{16} - 4 x^{15} + 28460 x^{14} + 90 x^{13} + 276395 x^{12} + 5430 x^{11} + 1727545 x^{10} + 61230 x^{9} + 6965605 x^{8} + 202090 x^{7} + 17624360 x^{6} - 246742 x^{5} + 26512155 x^{4} - 1889530 x^{3} + 22642685 x^{2} - 1174080 x + 10740479 \)

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:  \(6472892377132656250000000000000000=2^{16}\cdot 5^{22}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.04$
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:  $2, 5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{3}{14} a^{16} + \frac{1}{14} a^{15} + \frac{3}{14} a^{14} + \frac{1}{7} a^{12} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{5}{14} a^{8} - \frac{2}{7} a^{7} + \frac{3}{14} a^{6} - \frac{5}{14} a^{5} - \frac{2}{7} a^{4} - \frac{3}{14} a^{3} - \frac{3}{14} a^{2} + \frac{3}{7} a + \frac{1}{14}$, $\frac{1}{8004267130931940208479210885343260430757393425875644843698} a^{19} - \frac{102264640770677897230640272447960063157500253008808234364}{4002133565465970104239605442671630215378696712937822421849} a^{18} - \frac{587698686827607115560572376877994241510063530773835844251}{8004267130931940208479210885343260430757393425875644843698} a^{17} - \frac{574622254892657717793890280471298426296505255387890010487}{4002133565465970104239605442671630215378696712937822421849} a^{16} + \frac{122527566210154285846133958957642118467174746118151227549}{571733366495138586319943634667375745054099530419688917407} a^{15} - \frac{626404061639619580595636779668690191994664862640133178741}{8004267130931940208479210885343260430757393425875644843698} a^{14} - \frac{142322699186107167483377864945192402876745382748385268750}{4002133565465970104239605442671630215378696712937822421849} a^{13} + \frac{945645018908495359385073085815656686733318629214163868522}{4002133565465970104239605442671630215378696712937822421849} a^{12} + \frac{634609046700105163191316557569427008123735016933365960176}{4002133565465970104239605442671630215378696712937822421849} a^{11} + \frac{658478824000432036740525709058663140485623093363086999607}{8004267130931940208479210885343260430757393425875644843698} a^{10} + \frac{518152750568258330514651671819249136998368979779739048563}{1143466732990277172639887269334751490108199060839377834814} a^{9} + \frac{1667552550536169238471451535159603850792111429979174674365}{4002133565465970104239605442671630215378696712937822421849} a^{8} + \frac{981526840162404256675539185886281687852260001880807589425}{8004267130931940208479210885343260430757393425875644843698} a^{7} + \frac{145620140373372120536547369916067090842646610296468373413}{1143466732990277172639887269334751490108199060839377834814} a^{6} + \frac{951103922529987844641710234790401132049561785491563554135}{4002133565465970104239605442671630215378696712937822421849} a^{5} + \frac{645440626534726457670119673415561395569554375216489113521}{4002133565465970104239605442671630215378696712937822421849} a^{4} - \frac{1483358348883054736425482262149992387247415455313188184227}{4002133565465970104239605442671630215378696712937822421849} a^{3} + \frac{816247457786926239687122110766870551766277316239449455963}{4002133565465970104239605442671630215378696712937822421849} a^{2} + \frac{774740552634067925223579824351036994486857601655982379873}{8004267130931940208479210885343260430757393425875644843698} a + \frac{4177665331420159195327126133782292526113812733877229310}{26504195797787881485030499620341921956150309357204121999}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 207001385.12693426 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{5}, \sqrt{-23})\), 5.1.50000.1, 10.2.12500000000.1, 10.0.16090857500000000.2, 10.0.80454287500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$