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Properties

Label 20.0.35251755556...1856.7
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 11^{18}$
Root discriminant $42.40$
Ramified primes $2, 3, 11$
Class number $620$ (GRH)
Class group $[2, 310]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![652081, -438908, 765484, -352162, 383064, -145432, 138157, -42190, 36517, -10362, 10198, -3344, 1822, -100, 266, -144, 66, 4, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 + 4*x^17 + 66*x^16 - 144*x^15 + 266*x^14 - 100*x^13 + 1822*x^12 - 3344*x^11 + 10198*x^10 - 10362*x^9 + 36517*x^8 - 42190*x^7 + 138157*x^6 - 145432*x^5 + 383064*x^4 - 352162*x^3 + 765484*x^2 - 438908*x + 652081)
 
gp: K = bnfinit(x^20 - 2*x^19 + x^18 + 4*x^17 + 66*x^16 - 144*x^15 + 266*x^14 - 100*x^13 + 1822*x^12 - 3344*x^11 + 10198*x^10 - 10362*x^9 + 36517*x^8 - 42190*x^7 + 138157*x^6 - 145432*x^5 + 383064*x^4 - 352162*x^3 + 765484*x^2 - 438908*x + 652081, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + x^{18} + 4 x^{17} + 66 x^{16} - 144 x^{15} + 266 x^{14} - 100 x^{13} + 1822 x^{12} - 3344 x^{11} + 10198 x^{10} - 10362 x^{9} + 36517 x^{8} - 42190 x^{7} + 138157 x^{6} - 145432 x^{5} + 383064 x^{4} - 352162 x^{3} + 765484 x^{2} - 438908 x + 652081 \)

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:  \(352517555563337816067682238201856=2^{30}\cdot 3^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.40$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(83,·)$, $\chi_{264}(67,·)$, $\chi_{264}(65,·)$, $\chi_{264}(17,·)$, $\chi_{264}(131,·)$, $\chi_{264}(235,·)$, $\chi_{264}(25,·)$, $\chi_{264}(233,·)$, $\chi_{264}(97,·)$, $\chi_{264}(35,·)$, $\chi_{264}(163,·)$, $\chi_{264}(227,·)$, $\chi_{264}(161,·)$, $\chi_{264}(107,·)$, $\chi_{264}(49,·)$, $\chi_{264}(91,·)$, $\chi_{264}(115,·)$, $\chi_{264}(41,·)$, $\chi_{264}(169,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{331} a^{18} - \frac{161}{331} a^{17} - \frac{3}{331} a^{16} - \frac{41}{331} a^{15} - \frac{18}{331} a^{14} - \frac{139}{331} a^{13} - \frac{39}{331} a^{12} + \frac{48}{331} a^{11} + \frac{38}{331} a^{10} - \frac{59}{331} a^{9} - \frac{55}{331} a^{8} - \frac{69}{331} a^{7} - \frac{85}{331} a^{6} - \frac{149}{331} a^{5} - \frac{81}{331} a^{4} - \frac{81}{331} a^{3} - \frac{136}{331} a^{2} - \frac{72}{331} a - \frac{37}{331}$, $\frac{1}{7007015472022199661941310962274625726117380347} a^{19} + \frac{3254980906490592870848446017845988261803870}{7007015472022199661941310962274625726117380347} a^{18} + \frac{1284536508620675405542184327755503147399030309}{7007015472022199661941310962274625726117380347} a^{17} + \frac{2143288498257196704110441870661129184665723323}{7007015472022199661941310962274625726117380347} a^{16} - \frac{3482042778550830967268864646949014431646113925}{7007015472022199661941310962274625726117380347} a^{15} - \frac{2410142704827566736630625322389585292036453157}{7007015472022199661941310962274625726117380347} a^{14} + \frac{1523206434981943790900625264115146483402833559}{7007015472022199661941310962274625726117380347} a^{13} + \frac{473600426737310700378708922402956995827044027}{7007015472022199661941310962274625726117380347} a^{12} - \frac{244291982034390146938951122649417667011421841}{7007015472022199661941310962274625726117380347} a^{11} - \frac{1028701951241665496687759601706336226020691096}{7007015472022199661941310962274625726117380347} a^{10} - \frac{722112015143817363482046155806337361207435735}{7007015472022199661941310962274625726117380347} a^{9} + \frac{3497088815981119584672311027240274408938767556}{7007015472022199661941310962274625726117380347} a^{8} - \frac{1323319562591170859559916709945693136045235670}{7007015472022199661941310962274625726117380347} a^{7} - \frac{672108193431791588438606209487126075228230755}{7007015472022199661941310962274625726117380347} a^{6} + \frac{1423343848647379301802656754657842787843880722}{7007015472022199661941310962274625726117380347} a^{5} - \frac{3048769434957004320602960885999748443538984724}{7007015472022199661941310962274625726117380347} a^{4} + \frac{2327971105452456144243196699566006038383345607}{7007015472022199661941310962274625726117380347} a^{3} + \frac{568324175075680340224201789017074048362148988}{7007015472022199661941310962274625726117380347} a^{2} + \frac{2241897412023019724409898108166288732445822919}{7007015472022199661941310962274625726117380347} a + \frac{2253023209322269794467455429786210387151765638}{7007015472022199661941310962274625726117380347}$

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

Class group and class number

$C_{2}\times C_{310}$, which has order $620$ (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:  \( 125582.779517 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-66}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 10.0.18775450875101184.1, \(\Q(\zeta_{33})^+\), 10.0.7024111812608.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$