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Properties

Label 20.0.65416827983...000.12
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{34}\cdot 41^{18}$
Root discriminant $872.54$
Ramified primes $2, 5, 41$
Class number $15994880000$ (GRH)
Class group $[2, 4, 40, 40, 40, 31240]$ (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![17023928122802176, 0, 26398034793363200, 0, 4775088327115025, 0, 367779268319780, 0, 15206844638930, 0, 370384365328, 0, 5529189275, 0, 50790800, 0, 277570, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 277570*x^16 + 50790800*x^14 + 5529189275*x^12 + 370384365328*x^10 + 15206844638930*x^8 + 367779268319780*x^6 + 4775088327115025*x^4 + 26398034793363200*x^2 + 17023928122802176)
 
gp: K = bnfinit(x^20 + 820*x^18 + 277570*x^16 + 50790800*x^14 + 5529189275*x^12 + 370384365328*x^10 + 15206844638930*x^8 + 367779268319780*x^6 + 4775088327115025*x^4 + 26398034793363200*x^2 + 17023928122802176, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 277570 x^{16} + 50790800 x^{14} + 5529189275 x^{12} + 370384365328 x^{10} + 15206844638930 x^{8} + 367779268319780 x^{6} + 4775088327115025 x^{4} + 26398034793363200 x^{2} + 17023928122802176 \)

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:  \(65416827983112661350313274854125976562500000000000000000000=2^{20}\cdot 5^{34}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $872.54$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4100=2^{2}\cdot 5^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4100}(1,·)$, $\chi_{4100}(4099,·)$, $\chi_{4100}(2049,·)$, $\chi_{4100}(1419,·)$, $\chi_{4100}(461,·)$, $\chi_{4100}(2511,·)$, $\chi_{4100}(209,·)$, $\chi_{4100}(2259,·)$, $\chi_{4100}(1371,·)$, $\chi_{4100}(3469,·)$, $\chi_{4100}(3421,·)$, $\chi_{4100}(2051,·)$, $\chi_{4100}(679,·)$, $\chi_{4100}(2729,·)$, $\chi_{4100}(1841,·)$, $\chi_{4100}(3891,·)$, $\chi_{4100}(1589,·)$, $\chi_{4100}(3639,·)$, $\chi_{4100}(631,·)$, $\chi_{4100}(2681,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{5} + \frac{1}{8} a^{3} - \frac{3}{16} a$, $\frac{1}{1312} a^{10} + \frac{1}{16} a^{8} - \frac{1}{32} a^{6} - \frac{3}{16} a^{4} + \frac{1}{32} a^{2}$, $\frac{1}{1312} a^{11} + \frac{3}{32} a^{7} - \frac{1}{8} a^{5} - \frac{3}{32} a^{3} + \frac{3}{16} a$, $\frac{1}{1312} a^{12} + \frac{3}{32} a^{8} - \frac{1}{8} a^{6} - \frac{3}{32} a^{4} + \frac{3}{16} a^{2}$, $\frac{1}{40672} a^{13} - \frac{5}{40672} a^{11} - \frac{11}{992} a^{9} - \frac{23}{992} a^{7} - \frac{33}{992} a^{5} - \frac{199}{992} a^{3} + \frac{51}{248} a$, $\frac{1}{2765696} a^{14} - \frac{315}{2765696} a^{12} - \frac{427}{1382848} a^{10} + \frac{1217}{67456} a^{8} - \frac{1303}{33728} a^{6} - \frac{22395}{67456} a^{4} - \frac{8259}{67456} a^{2} + \frac{1}{17}$, $\frac{1}{1109044096} a^{15} + \frac{11245}{1109044096} a^{13} + \frac{13887}{554522048} a^{11} + \frac{409489}{27049856} a^{9} + \frac{277871}{13524928} a^{7} - \frac{1470523}{27049856} a^{5} + \frac{4152321}{27049856} a^{3} + \frac{781619}{1690616} a$, $\frac{1}{107577277312} a^{16} + \frac{803}{6328075136} a^{14} + \frac{237883}{1582018784} a^{12} - \frac{1967543}{6328075136} a^{10} + \frac{1567815}{38585824} a^{8} + \frac{17097245}{154343296} a^{6} - \frac{1234681}{154343296} a^{4} - \frac{31070367}{77171648} a^{2} + \frac{91}{1649}$, $\frac{1}{107577277312} a^{17} - \frac{13}{53788638656} a^{15} - \frac{82029}{107577277312} a^{13} - \frac{35078801}{107577277312} a^{11} + \frac{44651991}{2623836032} a^{9} + \frac{267683371}{2623836032} a^{7} + \frac{150746711}{1311918016} a^{5} + \frac{466540017}{2623836032} a^{3} - \frac{758465}{2411614} a$, $\frac{1}{386081159048390234393068072662884665488128} a^{18} + \frac{351584431051015808888984231059}{193040579524195117196534036331442332744064} a^{16} - \frac{5617104959345445806227726268417937}{48260144881048779299133509082860583186016} a^{14} - \frac{388606593843667853119765702702894325}{1556778867130605783843016422027760747936} a^{12} + \frac{66278790765295705653527208459163842573}{386081159048390234393068072662884665488128} a^{10} - \frac{75570081167272125636088025938754385795}{1177076704415823885344719733728306906976} a^{8} + \frac{66763402115663391334052363190193256735}{2354153408831647770689439467456613813952} a^{6} + \frac{1055866978958310017462209946285291134385}{4708306817663295541378878934913227627904} a^{4} + \frac{46631378056885998405701607059933608077}{553918449136858298985750462930967956224} a^{2} + \frac{1079290973676929778419791185713816}{2959025582253599583060292146972053}$, $\frac{1}{253269240335743993761852655666852340560211968} a^{19} + \frac{945272543327869863039317970365}{1544324636193560937572272290651538661952512} a^{17} - \frac{737038793675336968999918416915751}{3088649272387121875144544581303077323905024} a^{15} - \frac{2041694972090199377729986466817753397}{386081159048390234393068072662884665488128} a^{13} + \frac{764055624658817630476166679794293218883}{6177298544774243750289089162606154647810048} a^{11} + \frac{6630938812571772072061175842508498473371}{386081159048390234393068072662884665488128} a^{9} + \frac{7789720388944311200921273490686739033513}{75332909082612728662062062958611642046464} a^{7} - \frac{2145692510435739596262007295339832996855}{37666454541306364331031031479305821023232} a^{5} - \frac{15266822471826362308060477594729911493023}{150665818165225457324124125917223284092928} a^{3} + \frac{90772335271982815333815361876363710383}{294269176103955971336179933432076726744} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{40}\times C_{40}\times C_{40}\times C_{31240}$, which has order $15994880000$ (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:  \( \frac{216169635375}{1309349081739486928880021504} a^{19} + \frac{1038941328875}{7983835864265164200487936} a^{17} + \frac{21337701056937}{515086184791300916160512} a^{15} + \frac{13726572913295135}{1995958966066291050121984} a^{13} + \frac{20413739980524491885}{31935343457060656801951744} a^{11} + \frac{65318491252163753975}{1995958966066291050121984} a^{9} + \frac{313715328048057618775}{389455408012934839048192} a^{7} + \frac{820965983671091954167}{194727704006467419524096} a^{5} - \frac{104059068869195301423185}{778910816025869678096384} a^{3} - \frac{480978271369834228695}{380327546887631678758} a \) (order $4$)
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:  \( 3058814187510.6895 \) (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

\(\Q(\sqrt{-205}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{205}) \), \(\Q(i, \sqrt{205})\), 5.5.1103812890625.4, 10.0.255767136245282031250000000000.3, 10.0.1247644567050156250000000000.4, 10.10.249772593989533233642578125.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$5$5.10.17.3$x^{10} - 5 x^{8} + 30$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.3$x^{10} - 5 x^{8} + 30$$10$$1$$17$$C_{10}$$[2]_{2}$
$41$41.10.9.4$x^{10} - 1912896$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.4$x^{10} - 1912896$$10$$1$$9$$C_{10}$$[\ ]_{10}$