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Properties

Label 20.12.3972308189...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{16}\cdot 3^{2}\cdot 5^{22}\cdot 7^{10}$
Root discriminant $30.20$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T140

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -100, 495, 520, -3685, 720, 7060, -4790, -1825, 670, 481, 1470, -555, -650, 30, 118, 65, -10, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 - 10*x^17 + 65*x^16 + 118*x^15 + 30*x^14 - 650*x^13 - 555*x^12 + 1470*x^11 + 481*x^10 + 670*x^9 - 1825*x^8 - 4790*x^7 + 7060*x^6 + 720*x^5 - 3685*x^4 + 520*x^3 + 495*x^2 - 100*x - 5)
 
gp: K = bnfinit(x^20 - 15*x^18 - 10*x^17 + 65*x^16 + 118*x^15 + 30*x^14 - 650*x^13 - 555*x^12 + 1470*x^11 + 481*x^10 + 670*x^9 - 1825*x^8 - 4790*x^7 + 7060*x^6 + 720*x^5 - 3685*x^4 + 520*x^3 + 495*x^2 - 100*x - 5, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} - 10 x^{17} + 65 x^{16} + 118 x^{15} + 30 x^{14} - 650 x^{13} - 555 x^{12} + 1470 x^{11} + 481 x^{10} + 670 x^{9} - 1825 x^{8} - 4790 x^{7} + 7060 x^{6} + 720 x^{5} - 3685 x^{4} + 520 x^{3} + 495 x^{2} - 100 x - 5 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(397230818906250000000000000000=2^{16}\cdot 3^{2}\cdot 5^{22}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.20$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{14} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{11} - \frac{1}{14} a^{10} - \frac{2}{7} a^{9} - \frac{1}{14} a^{8} - \frac{3}{14} a^{6} - \frac{3}{7} a^{5} + \frac{3}{14} a^{4} - \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{5}{14}$, $\frac{1}{14} a^{15} + \frac{3}{14} a^{13} - \frac{1}{7} a^{12} + \frac{3}{14} a^{11} - \frac{1}{7} a^{10} - \frac{1}{2} a^{9} - \frac{5}{14} a^{8} - \frac{3}{14} a^{7} + \frac{1}{14} a^{5} + \frac{1}{14} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{3}{14}$, $\frac{1}{28} a^{16} + \frac{3}{14} a^{13} - \frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{5}{14} a^{6} - \frac{1}{14} a^{5} + \frac{5}{14} a^{4} + \frac{3}{7} a^{3} + \frac{3}{14} a^{2} - \frac{1}{2} a + \frac{1}{28}$, $\frac{1}{28} a^{17} - \frac{1}{14} a^{13} + \frac{1}{7} a^{12} - \frac{3}{14} a^{11} + \frac{3}{14} a^{10} + \frac{3}{28} a^{9} + \frac{3}{14} a^{8} - \frac{1}{7} a^{7} + \frac{1}{14} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{9}{28} a - \frac{3}{7}$, $\frac{1}{28} a^{18} - \frac{3}{14} a^{13} - \frac{3}{14} a^{12} + \frac{1}{14} a^{11} + \frac{1}{28} a^{10} - \frac{1}{14} a^{9} + \frac{2}{7} a^{8} + \frac{1}{14} a^{7} - \frac{1}{14} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{5}{28} a^{2} - \frac{5}{14} a + \frac{1}{7}$, $\frac{1}{45871349443286308775188628} a^{19} + \frac{21611055461710528128583}{45871349443286308775188628} a^{18} + \frac{6559526823779224667182}{1638262480117368170542451} a^{17} - \frac{361380421381596056909967}{22935674721643154387594314} a^{16} - \frac{197147134087758922051234}{11467837360821577193797157} a^{15} - \frac{404583261668546434722237}{22935674721643154387594314} a^{14} - \frac{1977086177207767113605183}{22935674721643154387594314} a^{13} - \frac{4705480265472761542855}{34698448898098569421474} a^{12} - \frac{85964553525391881435551}{6553049920469472682169804} a^{11} - \frac{3698442696539564907765133}{45871349443286308775188628} a^{10} + \frac{4282999752814766531229033}{22935674721643154387594314} a^{9} - \frac{9699389636270910483152305}{22935674721643154387594314} a^{8} + \frac{238730213663998085667026}{674578668283622187870421} a^{7} + \frac{20767203917932462765435}{1638262480117368170542451} a^{6} - \frac{1488363368669482168040383}{11467837360821577193797157} a^{5} - \frac{3102306102995992145406099}{22935674721643154387594314} a^{4} + \frac{702034118687013541645409}{3528565341791254521168356} a^{3} + \frac{246247665552411601545709}{504080763113036360166908} a^{2} + \frac{1897038329879733986171327}{11467837360821577193797157} a - \frac{2781532495641908856424961}{22935674721643154387594314}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 91555801.4786 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T140:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n140
Character table for t20n140 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.2450000.1, 10.6.126052500000000.1, 10.6.630262500000000.1, 10.10.30012500000000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 siblings: data not computed
Degree 40 siblings: 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 R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$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}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$