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Properties

Label 20.6.73931034575...5547.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,13^{10}\cdot 97^{4}\cdot 347^{7}$
Root discriminant $69.73$
Ramified primes $13, 97, 347$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4831544171, 1802762076, -352647068, -2029101662, 933901288, -7431976, 1297915, 40552439, -13876331, 14896738, 873214, -499338, 312014, -76948, -236, 1026, -518, 76, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 8*x^18 + 76*x^17 - 518*x^16 + 1026*x^15 - 236*x^14 - 76948*x^13 + 312014*x^12 - 499338*x^11 + 873214*x^10 + 14896738*x^9 - 13876331*x^8 + 40552439*x^7 + 1297915*x^6 - 7431976*x^5 + 933901288*x^4 - 2029101662*x^3 - 352647068*x^2 + 1802762076*x - 4831544171)
 
gp: K = bnfinit(x^20 - 4*x^19 - 8*x^18 + 76*x^17 - 518*x^16 + 1026*x^15 - 236*x^14 - 76948*x^13 + 312014*x^12 - 499338*x^11 + 873214*x^10 + 14896738*x^9 - 13876331*x^8 + 40552439*x^7 + 1297915*x^6 - 7431976*x^5 + 933901288*x^4 - 2029101662*x^3 - 352647068*x^2 + 1802762076*x - 4831544171, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 8 x^{18} + 76 x^{17} - 518 x^{16} + 1026 x^{15} - 236 x^{14} - 76948 x^{13} + 312014 x^{12} - 499338 x^{11} + 873214 x^{10} + 14896738 x^{9} - 13876331 x^{8} + 40552439 x^{7} + 1297915 x^{6} - 7431976 x^{5} + 933901288 x^{4} - 2029101662 x^{3} - 352647068 x^{2} + 1802762076 x - 4831544171 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7393103457599271381703858996552825547=-\,13^{10}\cdot 97^{4}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.73$
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:  $13, 97, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{19} + \frac{758629478302785208041292841571564778814311613200034514748806846874405713841256108912626319455849656031}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{18} + \frac{106415805198787703695581630892091594013977859725439764011405436145990296549704751491950161988825611514}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{17} + \frac{519527653276921520396470399764099136702751637757590752512555916725537423234704610904644755734031184929}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{16} + \frac{801636743111269860876035652151573727618489237401225031462625316479727791227595353633010562425317836766}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{15} - \frac{1037417529595545179308619622986139699831234343496822144791852134070750920515708022718562565525795715430}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{14} - \frac{975122710319544032238092074169463348357619503262804553105252095137923180238216977628986437916277204002}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{13} - \frac{1220157837331291083297967702122574923683182636337423579718980976579285501386221059752354433921920669174}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{12} - \frac{41020723579428769523547934800274244009840035892210114427665209751764377797517813836045361348550568964}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{11} + \frac{469860560614462162184475875314700691928769407932793469714387812282652740839659645730164997152187901678}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{10} - \frac{528130288981778774839613337195277118792779136143939237949592008741395100401276426651582663349901686313}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{9} + \frac{365969979606559341657728719633482072791238719218278096354559965482177126305277114915423814940067972165}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{8} + \frac{649086977274695019597864267788657407491509748463134016107147593441057683408085415129552965313996713017}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{7} - \frac{381393578973858126053766834055092817517346532071864855100819195674688278894968830439717637634363281459}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{6} - \frac{1223566831413988087911238143056115949950600512484608959489849898385059635112015958260507575010609343563}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{5} - \frac{99424777159212042195647276734177766388992164159169877215157413624424227553077483880387032999333671697}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{4} + \frac{1213808289435105591135452588553196057667042736608348854571373653197804981017171868509077601830006344259}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{3} - \frac{54772231459812931627776383685582224929123377823285336372762160584136666072703712280585467069800591952}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a^{2} - \frac{399472042650066360604467795837539727441998056726971997117529501797807134282044182001059443147438475269}{2657138890600227796430520835709644674763394103559127764461547413492096521454498846415789762994426865311} a - \frac{12648148014957352912109105788891462459671639922400368615170871831449341640934711934059914565213478583}{26308305847527007885450701343659848264984100035238888757045023895961351697569295509067225376182444211}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $12$
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:  \( 19456377296.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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

Sibling fields

Degree 20 sibling: 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
347Data not computed