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Properties

Label 20.0.17654703062...0000.7
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{34}\cdot 7^{10}$
Root discriminant $115.44$
Ramified primes $2, 5, 7$
Class number $13629440$ (GRH)
Class group $[4, 4, 44, 44, 440]$ (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![1057144858799, -2457333470, 517958259535, -303834980, 120490616165, 4691704, 17501128105, 3441980, 1756855370, 298470, 127430747, 11580, 6773795, 170, 261280, -2, 7030, 0, 120, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 120*x^18 + 7030*x^16 - 2*x^15 + 261280*x^14 + 170*x^13 + 6773795*x^12 + 11580*x^11 + 127430747*x^10 + 298470*x^9 + 1756855370*x^8 + 3441980*x^7 + 17501128105*x^6 + 4691704*x^5 + 120490616165*x^4 - 303834980*x^3 + 517958259535*x^2 - 2457333470*x + 1057144858799)
 
gp: K = bnfinit(x^20 + 120*x^18 + 7030*x^16 - 2*x^15 + 261280*x^14 + 170*x^13 + 6773795*x^12 + 11580*x^11 + 127430747*x^10 + 298470*x^9 + 1756855370*x^8 + 3441980*x^7 + 17501128105*x^6 + 4691704*x^5 + 120490616165*x^4 - 303834980*x^3 + 517958259535*x^2 - 2457333470*x + 1057144858799, 1)
 

Normalized defining polynomial

\( x^{20} + 120 x^{18} + 7030 x^{16} - 2 x^{15} + 261280 x^{14} + 170 x^{13} + 6773795 x^{12} + 11580 x^{11} + 127430747 x^{10} + 298470 x^{9} + 1756855370 x^{8} + 3441980 x^{7} + 17501128105 x^{6} + 4691704 x^{5} + 120490616165 x^{4} - 303834980 x^{3} + 517958259535 x^{2} - 2457333470 x + 1057144858799 \)

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:  \(176547030625000000000000000000000000000000=2^{30}\cdot 5^{34}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.44$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1400=2^{3}\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1400}(1,·)$, $\chi_{1400}(69,·)$, $\chi_{1400}(449,·)$, $\chi_{1400}(1289,·)$, $\chi_{1400}(909,·)$, $\chi_{1400}(461,·)$, $\chi_{1400}(1301,·)$, $\chi_{1400}(281,·)$, $\chi_{1400}(729,·)$, $\chi_{1400}(349,·)$, $\chi_{1400}(741,·)$, $\chi_{1400}(1121,·)$, $\chi_{1400}(1189,·)$, $\chi_{1400}(561,·)$, $\chi_{1400}(169,·)$, $\chi_{1400}(1009,·)$, $\chi_{1400}(629,·)$, $\chi_{1400}(841,·)$, $\chi_{1400}(1021,·)$, $\chi_{1400}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{19} + \frac{45509000120495153607853922715055986495736759308631982246986035408557269445026205}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{18} + \frac{4384259934163007155973659799805380972214323213023576217836004283120330481992380}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{17} + \frac{21799753466856989309888443956307035891353076566164115162361510680908243608625009}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{16} - \frac{22518019823546954959381408458753923135606556830008871415920515775369763267626862}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{15} + \frac{59047944980520960803185453874164451206858464167165790750548993165506378844717363}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{14} + \frac{25850187405005501994229581067428635269063485124561287153330568627852468727415679}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{13} - \frac{243937934049778739619120994295472586165895242665946099991224883806502752707648}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{12} + \frac{60977453535226450052280466224373398852359660887822382578345326398312119415011486}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{11} - \frac{58469783456438601705657386669533297243174053600288242611015665361754250841451482}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{10} + \frac{25336438776455239381124136830377865238826441763800774259077168005141024334674001}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{9} - \frac{33465725106325302242086890524578244226021107960727299033171547608807654334713994}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{8} - \frac{23407482786970212269231326227102159965122812996323113470034413368714036545537799}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{7} - \frac{2894735609287169584413278300834067487606726924927678025541160124998766322246344}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{6} + \frac{38925548171405131480619086372548658927631426130599893502438834040666637275910303}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{5} - \frac{57855726313145363954610961037981083542702091425207020171310979346764683193536947}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{4} - \frac{47433253897321430270381597880265743110287047988699801247726363858023660915159494}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{3} + \frac{252223392808090906234273298758371105802245625727936408020454661792731787169437}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a^{2} - \frac{24243447674536152356528812109700022981841744362783103748822683749551849246718581}{128550635971304673017338293389361322608923132629420010067773617435488895247745301} a + \frac{29393033236114996583122483034857335370200043597124148314117283939016930000368058}{128550635971304673017338293389361322608923132629420010067773617435488895247745301}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{44}\times C_{44}\times C_{440}$, which has order $13629440$ (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:  \( 161406.8376411007 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{5}, \sqrt{-14})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.420175000000000000000.3, 10.0.84035000000000000000.3

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 ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$