Properties

Label 33.33.277...001.1
Degree $33$
Signature $[33, 0]$
Discriminant $2.780\times 10^{59}$
Root discriminant $63.29$
Ramified primes $7, 23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{33}$ (as 33T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^33 - x^32 - 52*x^31 + 47*x^30 + 1159*x^29 - 945*x^28 - 14589*x^27 + 10741*x^26 + 115099*x^25 - 76770*x^24 - 597580*x^23 + 362822*x^22 + 2088771*x^21 - 1160546*x^20 - 4956062*x^19 + 2531215*x^18 + 7980244*x^17 - 3753538*x^16 - 8674935*x^15 + 3750773*x^14 + 6304834*x^13 - 2497276*x^12 - 3006793*x^11 + 1087323*x^10 + 908614*x^9 - 298518*x^8 - 163557*x^7 + 48112*x^6 + 15777*x^5 - 3955*x^4 - 691*x^3 + 126*x^2 + 12*x - 1)
 
gp: K = bnfinit(x^33 - x^32 - 52*x^31 + 47*x^30 + 1159*x^29 - 945*x^28 - 14589*x^27 + 10741*x^26 + 115099*x^25 - 76770*x^24 - 597580*x^23 + 362822*x^22 + 2088771*x^21 - 1160546*x^20 - 4956062*x^19 + 2531215*x^18 + 7980244*x^17 - 3753538*x^16 - 8674935*x^15 + 3750773*x^14 + 6304834*x^13 - 2497276*x^12 - 3006793*x^11 + 1087323*x^10 + 908614*x^9 - 298518*x^8 - 163557*x^7 + 48112*x^6 + 15777*x^5 - 3955*x^4 - 691*x^3 + 126*x^2 + 12*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 12, 126, -691, -3955, 15777, 48112, -163557, -298518, 908614, 1087323, -3006793, -2497276, 6304834, 3750773, -8674935, -3753538, 7980244, 2531215, -4956062, -1160546, 2088771, 362822, -597580, -76770, 115099, 10741, -14589, -945, 1159, 47, -52, -1, 1]);
 

\( x^{33} - x^{32} - 52 x^{31} + 47 x^{30} + 1159 x^{29} - 945 x^{28} - 14589 x^{27} + 10741 x^{26} + 115099 x^{25} - 76770 x^{24} - 597580 x^{23} + 362822 x^{22} + 2088771 x^{21} - 1160546 x^{20} - 4956062 x^{19} + 2531215 x^{18} + 7980244 x^{17} - 3753538 x^{16} - 8674935 x^{15} + 3750773 x^{14} + 6304834 x^{13} - 2497276 x^{12} - 3006793 x^{11} + 1087323 x^{10} + 908614 x^{9} - 298518 x^{8} - 163557 x^{7} + 48112 x^{6} + 15777 x^{5} - 3955 x^{4} - 691 x^{3} + 126 x^{2} + 12 x - 1 \)

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

Invariants

Degree:  $33$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[33, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(277966181338944111003326058293667039541136678070715028736001\)\(\medspace = 7^{22}\cdot 23^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $63.29$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $33$
This field is Galois and abelian over $\Q$.
Conductor:  \(161=7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{161}(128,·)$, $\chi_{161}(1,·)$, $\chi_{161}(2,·)$, $\chi_{161}(4,·)$, $\chi_{161}(8,·)$, $\chi_{161}(9,·)$, $\chi_{161}(141,·)$, $\chi_{161}(142,·)$, $\chi_{161}(16,·)$, $\chi_{161}(18,·)$, $\chi_{161}(151,·)$, $\chi_{161}(25,·)$, $\chi_{161}(156,·)$, $\chi_{161}(29,·)$, $\chi_{161}(32,·)$, $\chi_{161}(36,·)$, $\chi_{161}(39,·)$, $\chi_{161}(50,·)$, $\chi_{161}(58,·)$, $\chi_{161}(64,·)$, $\chi_{161}(71,·)$, $\chi_{161}(72,·)$, $\chi_{161}(78,·)$, $\chi_{161}(81,·)$, $\chi_{161}(85,·)$, $\chi_{161}(93,·)$, $\chi_{161}(95,·)$, $\chi_{161}(144,·)$, $\chi_{161}(100,·)$, $\chi_{161}(116,·)$, $\chi_{161}(121,·)$, $\chi_{161}(123,·)$, $\chi_{161}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{461} a^{31} + \frac{83}{461} a^{30} + \frac{141}{461} a^{29} + \frac{129}{461} a^{28} - \frac{177}{461} a^{27} - \frac{113}{461} a^{26} - \frac{209}{461} a^{25} - \frac{55}{461} a^{24} - \frac{3}{461} a^{23} - \frac{139}{461} a^{22} - \frac{222}{461} a^{21} - \frac{196}{461} a^{20} - \frac{115}{461} a^{19} - \frac{106}{461} a^{18} + \frac{38}{461} a^{17} + \frac{165}{461} a^{16} - \frac{225}{461} a^{14} + \frac{164}{461} a^{13} - \frac{151}{461} a^{12} + \frac{137}{461} a^{11} + \frac{153}{461} a^{10} - \frac{15}{461} a^{9} + \frac{10}{461} a^{8} + \frac{166}{461} a^{7} - \frac{160}{461} a^{6} + \frac{14}{461} a^{5} - \frac{132}{461} a^{4} + \frac{139}{461} a^{3} - \frac{89}{461} a^{2} + \frac{134}{461} a + \frac{200}{461}$, $\frac{1}{31395026281391975704557151272424583190591105677389094293} a^{32} - \frac{26641326808036072616049631227199038607447162638377869}{31395026281391975704557151272424583190591105677389094293} a^{31} - \frac{12394667625603363546226793672614844815981920740946199780}{31395026281391975704557151272424583190591105677389094293} a^{30} - \frac{9293397518544433009949847783767558410910687354746423199}{31395026281391975704557151272424583190591105677389094293} a^{29} + \frac{10716673645843509721343661028264448299963543435839704740}{31395026281391975704557151272424583190591105677389094293} a^{28} - \frac{2822536360795372717826849460698247405836227652781192025}{31395026281391975704557151272424583190591105677389094293} a^{27} + \frac{5083785012421412648559570904931865107270857925032766313}{31395026281391975704557151272424583190591105677389094293} a^{26} - \frac{10049864308704031228147647934193061176826257350333620312}{31395026281391975704557151272424583190591105677389094293} a^{25} + \frac{13654837512647334999052004398196497036981817619893144051}{31395026281391975704557151272424583190591105677389094293} a^{24} - \frac{2390685812685780397522832062865741183236718202271663116}{31395026281391975704557151272424583190591105677389094293} a^{23} - \frac{4652663215676084163757583408058855907902012924679103850}{31395026281391975704557151272424583190591105677389094293} a^{22} - \frac{7143855900703482539674674793874529354623152387383061784}{31395026281391975704557151272424583190591105677389094293} a^{21} + \frac{14718205999005488005461225459310857548507984435615394631}{31395026281391975704557151272424583190591105677389094293} a^{20} - \frac{15120621614547424115904312714180677415092768673595795353}{31395026281391975704557151272424583190591105677389094293} a^{19} + \frac{12974746979603050541343265461226947263334458320301317475}{31395026281391975704557151272424583190591105677389094293} a^{18} + \frac{8575196174630180370096381350798907522749009498376244847}{31395026281391975704557151272424583190591105677389094293} a^{17} + \frac{2072303481854416066564726447508029268418689264453124}{68102009287184329077130479983567425576119535091950313} a^{16} - \frac{12462939514904066101296816267378891132528714668459599876}{31395026281391975704557151272424583190591105677389094293} a^{15} - \frac{3669813908921967893511403232549472706608316644832168303}{31395026281391975704557151272424583190591105677389094293} a^{14} + \frac{13293398415629190125073647914263350063814137073049200259}{31395026281391975704557151272424583190591105677389094293} a^{13} + \frac{5322506342122152571590422454843116060573771433122128065}{31395026281391975704557151272424583190591105677389094293} a^{12} - \frac{13637079123184912959171279410590323062650045924299630677}{31395026281391975704557151272424583190591105677389094293} a^{11} - \frac{5914787815788135750450793526810150039918498731293829347}{31395026281391975704557151272424583190591105677389094293} a^{10} - \frac{5160955925388286885532997087999528558369929699508787}{48825857358307893786247513642962026734978391411180551} a^{9} + \frac{13334565860509878707763189729762633364366826013169022572}{31395026281391975704557151272424583190591105677389094293} a^{8} + \frac{6778690865892286579757143248581348899540790003281995884}{31395026281391975704557151272424583190591105677389094293} a^{7} - \frac{2018886506653908208362999431905507472001013115806664163}{31395026281391975704557151272424583190591105677389094293} a^{6} - \frac{10889302049371313486630059451555245962703757131876887469}{31395026281391975704557151272424583190591105677389094293} a^{5} - \frac{7685361073419045545662686121602625718686102674650395685}{31395026281391975704557151272424583190591105677389094293} a^{4} + \frac{9077328101109960918558630536765414991800034601564046768}{31395026281391975704557151272424583190591105677389094293} a^{3} + \frac{15167723390332439012558712650380335723095088667114394662}{31395026281391975704557151272424583190591105677389094293} a^{2} + \frac{974297440055815695388646681027201204182409299874924670}{31395026281391975704557151272424583190591105677389094293} a + \frac{7254656725807878303149125086224817068514570854873346}{68102009287184329077130479983567425576119535091950313}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $32$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 15716532070591353000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{33}\cdot(2\pi)^{0}\cdot 15716532070591353000 \cdot 1}{2\sqrt{277966181338944111003326058293667039541136678070715028736001}}\approx 0.128032611487433$ (assuming GRH)

Galois group

$C_{33}$ (as 33T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 33
The 33 conjugacy class representatives for $C_{33}$
Character table for $C_{33}$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), \(\Q(\zeta_{23})^+\)

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 $33$ $33$ $33$ R $33$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{3}$ $33$ $33$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{3}$ $33$ $33$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{11}$ $33$ $33$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
23Data not computed