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Properties

Label 20.0.13610161416...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $113.94$
Ramified primes $2, 5, 7, 11$
Class number $1362020$ (GRH)
Class group $[2, 681010]$ (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![61511627759, -20527319894, 37784285875, -10764612716, 10452990189, -2620416334, 1768558421, -399713644, 208321354, -42959640, 18238950, -3434802, 1225192, -207826, 63172, -9340, 2435, -302, 67, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 67*x^18 - 302*x^17 + 2435*x^16 - 9340*x^15 + 63172*x^14 - 207826*x^13 + 1225192*x^12 - 3434802*x^11 + 18238950*x^10 - 42959640*x^9 + 208321354*x^8 - 399713644*x^7 + 1768558421*x^6 - 2620416334*x^5 + 10452990189*x^4 - 10764612716*x^3 + 37784285875*x^2 - 20527319894*x + 61511627759)
 
gp: K = bnfinit(x^20 - 6*x^19 + 67*x^18 - 302*x^17 + 2435*x^16 - 9340*x^15 + 63172*x^14 - 207826*x^13 + 1225192*x^12 - 3434802*x^11 + 18238950*x^10 - 42959640*x^9 + 208321354*x^8 - 399713644*x^7 + 1768558421*x^6 - 2620416334*x^5 + 10452990189*x^4 - 10764612716*x^3 + 37784285875*x^2 - 20527319894*x + 61511627759, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 67 x^{18} - 302 x^{17} + 2435 x^{16} - 9340 x^{15} + 63172 x^{14} - 207826 x^{13} + 1225192 x^{12} - 3434802 x^{11} + 18238950 x^{10} - 42959640 x^{9} + 208321354 x^{8} - 399713644 x^{7} + 1768558421 x^{6} - 2620416334 x^{5} + 10452990189 x^{4} - 10764612716 x^{3} + 37784285875 x^{2} - 20527319894 x + 61511627759 \)

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:  \(136101614161182402364761324912640000000000=2^{30}\cdot 5^{10}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.94$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3080=2^{3}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3080}(1,·)$, $\chi_{3080}(1541,·)$, $\chi_{3080}(1609,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(141,·)$, $\chi_{3080}(421,·)$, $\chi_{3080}(2381,·)$, $\chi_{3080}(2729,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(1049,·)$, $\chi_{3080}(2589,·)$, $\chi_{3080}(69,·)$, $\chi_{3080}(1189,·)$, $\chi_{3080}(489,·)$, $\chi_{3080}(2029,·)$, $\chi_{3080}(1329,·)$, $\chi_{3080}(2869,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(1401,·)$, $\chi_{3080}(2941,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{43} a^{15} + \frac{17}{43} a^{14} + \frac{10}{43} a^{13} - \frac{5}{43} a^{12} - \frac{17}{43} a^{11} - \frac{12}{43} a^{10} - \frac{9}{43} a^{9} + \frac{14}{43} a^{8} + \frac{16}{43} a^{7} + \frac{18}{43} a^{6} + \frac{14}{43} a^{5} - \frac{8}{43} a^{4} - \frac{15}{43} a^{3} + \frac{4}{43} a^{2} + \frac{14}{43} a + \frac{1}{43}$, $\frac{1}{43} a^{16} - \frac{21}{43} a^{14} - \frac{3}{43} a^{13} - \frac{18}{43} a^{12} + \frac{19}{43} a^{11} - \frac{20}{43} a^{10} - \frac{5}{43} a^{9} - \frac{7}{43} a^{8} + \frac{4}{43} a^{7} + \frac{9}{43} a^{6} + \frac{12}{43} a^{5} - \frac{8}{43} a^{4} + \frac{1}{43} a^{3} - \frac{11}{43} a^{2} + \frac{21}{43} a - \frac{17}{43}$, $\frac{1}{43} a^{17} + \frac{10}{43} a^{14} + \frac{20}{43} a^{13} + \frac{10}{43} a^{11} + \frac{1}{43} a^{10} + \frac{19}{43} a^{9} - \frac{3}{43} a^{8} + \frac{1}{43} a^{7} + \frac{3}{43} a^{6} - \frac{15}{43} a^{5} + \frac{5}{43} a^{4} + \frac{18}{43} a^{3} + \frac{19}{43} a^{2} + \frac{19}{43} a + \frac{21}{43}$, $\frac{1}{150425776319030961859769} a^{18} + \frac{432523613910557643143}{150425776319030961859769} a^{17} - \frac{1023764676155428173874}{150425776319030961859769} a^{16} - \frac{1740548513890477452460}{150425776319030961859769} a^{15} - \frac{20688928257657795540377}{150425776319030961859769} a^{14} - \frac{23443132972330127914747}{150425776319030961859769} a^{13} - \frac{30372381038379163250889}{150425776319030961859769} a^{12} + \frac{31999602766326631817350}{150425776319030961859769} a^{11} + \frac{10409546278887876371720}{150425776319030961859769} a^{10} + \frac{33692550140539227253297}{150425776319030961859769} a^{9} + \frac{65002915159591248640139}{150425776319030961859769} a^{8} + \frac{8778659304530415216285}{150425776319030961859769} a^{7} + \frac{23274881252260128497416}{150425776319030961859769} a^{6} + \frac{67734774389018638960684}{150425776319030961859769} a^{5} + \frac{48662701641673829674492}{150425776319030961859769} a^{4} + \frac{12564316557737961451202}{150425776319030961859769} a^{3} - \frac{25651840174684310394320}{150425776319030961859769} a^{2} - \frac{44077803633281393733}{152098863821062651021} a + \frac{8637398615849266009262}{150425776319030961859769}$, $\frac{1}{258756470684492258957096477179242369345047515716135222161} a^{19} - \frac{469149347704586181102944151058184}{258756470684492258957096477179242369345047515716135222161} a^{18} - \frac{335386401052530112317737279282549973771613221505640796}{258756470684492258957096477179242369345047515716135222161} a^{17} + \frac{330841679751313402797254995510327133340189875160326208}{258756470684492258957096477179242369345047515716135222161} a^{16} + \frac{3292025482977134544145966290033253611076567991767809}{11250281334108359085091151181706189971523805031136314007} a^{15} + \frac{13812050813164141864786811707474598122050534033684960346}{258756470684492258957096477179242369345047515716135222161} a^{14} + \frac{84250038545402936801515320920535091707530662095024400848}{258756470684492258957096477179242369345047515716135222161} a^{13} - \frac{4878562192702872711489476934347839224027329720433905535}{258756470684492258957096477179242369345047515716135222161} a^{12} + \frac{179501650916984653172432855632162235618834812424351909}{11250281334108359085091151181706189971523805031136314007} a^{11} + \frac{48893974620157707517180078983424476601622604410387239465}{258756470684492258957096477179242369345047515716135222161} a^{10} + \frac{52086709808267583533638195724218566480546811374184438598}{258756470684492258957096477179242369345047515716135222161} a^{9} + \frac{94598292110666292754791500684567487451483236652577475402}{258756470684492258957096477179242369345047515716135222161} a^{8} - \frac{100149765927627146308596941010971542641111541739535153891}{258756470684492258957096477179242369345047515716135222161} a^{7} + \frac{11545695821135726468809589827409893094320987138809662551}{258756470684492258957096477179242369345047515716135222161} a^{6} + \frac{8103405244914585556901313955327452532299656421907860534}{258756470684492258957096477179242369345047515716135222161} a^{5} - \frac{15271540306444526579760845552143601967983229634583681267}{258756470684492258957096477179242369345047515716135222161} a^{4} - \frac{46840512200742928213370166851085683684532724965800193752}{258756470684492258957096477179242369345047515716135222161} a^{3} + \frac{74658789440843854747581403046394939122945755538373823918}{258756470684492258957096477179242369345047515716135222161} a^{2} - \frac{113306520134559852299421893837249253206047140909939765724}{258756470684492258957096477179242369345047515716135222161} a - \frac{169764758439427408499010960898698945170634737596576997}{733021163412159373816137329119666768682854152170354737}$

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

Class group and class number

$C_{2}\times C_{681010}$, which has order $1362020$ (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:  \( 530208.2507325789 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1, 10.0.368919522607820800000.1, 10.0.11258530353021875.4

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 R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$