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Properties

Label 20.0.35251755556...000.11
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $134.08$
Ramified primes $2, 3, 5, 11$
Class number $4792880$ (GRH)
Class group $[2, 2, 1198220]$ (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![22163170129, 0, 17367927780, 0, 6120549669, 0, 1270626240, 0, 170472586, 0, 15155944, 0, 869610, 0, 29472, 0, 565, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 + 565*x^16 + 29472*x^14 + 869610*x^12 + 15155944*x^10 + 170472586*x^8 + 1270626240*x^6 + 6120549669*x^4 + 17367927780*x^2 + 22163170129)
 
gp: K = bnfinit(x^20 + 20*x^18 + 565*x^16 + 29472*x^14 + 869610*x^12 + 15155944*x^10 + 170472586*x^8 + 1270626240*x^6 + 6120549669*x^4 + 17367927780*x^2 + 22163170129, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} + 565 x^{16} + 29472 x^{14} + 869610 x^{12} + 15155944 x^{10} + 170472586 x^{8} + 1270626240 x^{6} + 6120549669 x^{4} + 17367927780 x^{2} + 22163170129 \)

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:  \(3525175555633378160676822382018560000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.08$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(389,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(269,·)$, $\chi_{1320}(211,·)$, $\chi_{1320}(1109,·)$, $\chi_{1320}(1051,·)$, $\chi_{1320}(479,·)$, $\chi_{1320}(359,·)$, $\chi_{1320}(1319,·)$, $\chi_{1320}(509,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(811,·)$, $\chi_{1320}(749,·)$, $\chi_{1320}(239,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(571,·)$, $\chi_{1320}(931,·)$, $\chi_{1320}(959,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{4} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} - \frac{3}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} - \frac{3}{16} a$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{6} - \frac{3}{16} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{7} - \frac{3}{16} a^{3}$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{8} - \frac{5}{64} a^{4} - \frac{3}{64}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{9} - \frac{5}{64} a^{5} - \frac{3}{64} a$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{10} - \frac{5}{64} a^{6} - \frac{3}{64} a^{2}$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{11} - \frac{5}{64} a^{7} - \frac{3}{64} a^{3}$, $\frac{1}{256} a^{16} - \frac{3}{128} a^{8} - \frac{1}{32} a^{4} - \frac{3}{256}$, $\frac{1}{256} a^{17} - \frac{3}{128} a^{9} - \frac{1}{32} a^{5} - \frac{3}{256} a$, $\frac{1}{18368320056708035522301220096} a^{18} + \frac{3294822432575180717732357}{18368320056708035522301220096} a^{16} + \frac{3776828420365165816257869}{2296040007088504440287652512} a^{14} + \frac{9495797369074024367105015}{4592080014177008880575305024} a^{12} - \frac{169773933600762703576537911}{9184160028354017761150610048} a^{10} - \frac{205248569586050902647228429}{9184160028354017761150610048} a^{8} - \frac{35308771851755250206199125}{1148020003544252220143826256} a^{6} + \frac{467275185353200994784523363}{4592080014177008880575305024} a^{4} - \frac{5970179305688270506660613499}{18368320056708035522301220096} a^{2} - \frac{5841014831145036702541386371}{18368320056708035522301220096}$, $\frac{1}{2734546911802295372311549539351808} a^{19} + \frac{1574259952896382902344460114289}{1367273455901147686155774769675904} a^{17} - \frac{3756959659191946175852797896163}{683636727950573843077887384837952} a^{15} - \frac{1731778679549135400062594802161}{683636727950573843077887384837952} a^{13} + \frac{39851192111598469586676582264899}{1367273455901147686155774769675904} a^{11} + \frac{1058082312686851834369090802505}{341818363975286921538943692418976} a^{9} + \frac{14695447576531901101769018139133}{683636727950573843077887384837952} a^{7} - \frac{31247632703786091010627540316983}{683636727950573843077887384837952} a^{5} + \frac{547111376004795427481482301383137}{2734546911802295372311549539351808} a^{3} - \frac{542356665313685145956499064124261}{1367273455901147686155774769675904} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1198220}$, which has order $4792880$ (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:  \( 1589230.0087159988 \) (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

\(\Q(\sqrt{-165}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{22}, \sqrt{-30})\), \(\Q(\zeta_{11})^+\), 10.0.1833540124521600000.1, 10.10.77265229938688.1, 10.0.5333934907699200000.1

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 R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$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}$
5Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$