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Properties

Label 42.0.959...104.1
Degree $42$
Signature $[0, 21]$
Discriminant $-9.594\times 10^{77}$
Root discriminant \(71.90\)
Ramified primes $2,43$
Class number $3756652$ (GRH)
Class group [2, 1878326] (GRH)
Galois group $C_{42}$ (as 42T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1)
 
gp: K = bnfinit(y^42 + 41*y^40 + 780*y^38 + 9139*y^36 + 73815*y^34 + 435897*y^32 + 1947792*y^30 + 6724520*y^28 + 18156204*y^26 + 38567100*y^24 + 64512240*y^22 + 84672315*y^20 + 86493225*y^18 + 67863915*y^16 + 40116600*y^14 + 17383860*y^12 + 5311735*y^10 + 1081575*y^8 + 134596*y^6 + 8855*y^4 + 231*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1)
 

\( x^{42} + 41 x^{40} + 780 x^{38} + 9139 x^{36} + 73815 x^{34} + 435897 x^{32} + 1947792 x^{30} + 6724520 x^{28} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-959396304051793463814262846982490027578741814649477038563926538598268329263104\) \(\medspace = -\,2^{42}\cdot 43^{40}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(71.90\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 43^{20/21}\approx 71.89757443060533$
Ramified primes:   \(2\), \(43\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-1}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(172=2^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{172}(1,·)$, $\chi_{172}(17,·)$, $\chi_{172}(133,·)$, $\chi_{172}(135,·)$, $\chi_{172}(9,·)$, $\chi_{172}(11,·)$, $\chi_{172}(13,·)$, $\chi_{172}(15,·)$, $\chi_{172}(145,·)$, $\chi_{172}(21,·)$, $\chi_{172}(23,·)$, $\chi_{172}(25,·)$, $\chi_{172}(31,·)$, $\chi_{172}(35,·)$, $\chi_{172}(165,·)$, $\chi_{172}(167,·)$, $\chi_{172}(41,·)$, $\chi_{172}(47,·)$, $\chi_{172}(49,·)$, $\chi_{172}(53,·)$, $\chi_{172}(57,·)$, $\chi_{172}(59,·)$, $\chi_{172}(67,·)$, $\chi_{172}(79,·)$, $\chi_{172}(81,·)$, $\chi_{172}(83,·)$, $\chi_{172}(139,·)$, $\chi_{172}(87,·)$, $\chi_{172}(143,·)$, $\chi_{172}(95,·)$, $\chi_{172}(97,·)$, $\chi_{172}(99,·)$, $\chi_{172}(101,·)$, $\chi_{172}(103,·)$, $\chi_{172}(107,·)$, $\chi_{172}(109,·)$, $\chi_{172}(111,·)$, $\chi_{172}(117,·)$, $\chi_{172}(169,·)$, $\chi_{172}(153,·)$, $\chi_{172}(121,·)$, $\chi_{172}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{1048576}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{1878326}$, which has order $3756652$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -a^{41} - 40 a^{39} - 741 a^{37} - 8436 a^{35} - 66045 a^{33} - 376992 a^{31} - 1623160 a^{29} - 5379616 a^{27} - 13884156 a^{25} - 28048800 a^{23} - 44352165 a^{21} - 54627300 a^{19} - 51895935 a^{17} - 37442160 a^{15} - 20058300 a^{13} - 7726160 a^{11} - 2042975 a^{9} - 346104 a^{7} - 33649 a^{5} - 1540 a^{3} - 21 a \)  (order $4$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a^{39}+39a^{37}+702a^{35}+7735a^{33}+58344a^{31}+319176a^{29}+1308944a^{27}+4102137a^{25}+9924525a^{23}+18599295a^{21}+26936910a^{19}+29910465a^{17}+25110020a^{15}+15600900a^{13}+6953544a^{11}+2124694a^{9}+415701a^{7}+46683a^{5}+2470a^{3}+39a$, $a^{30}+30a^{28}+405a^{26}+3250a^{24}+17250a^{22}+63756a^{20}+168245a^{18}+319770a^{16}+436050a^{14}+419900a^{12}+277134a^{10}+119340a^{8}+30940a^{6}+4200a^{4}+225a^{2}+2$, $a^{26}+26a^{24}+299a^{22}+2002a^{20}+8645a^{18}+25194a^{16}+50388a^{14}+68952a^{12}+63206a^{10}+37180a^{8}+13013a^{6}+2366a^{4}+169a^{2}+1$, $a^{28}+28a^{26}+350a^{24}+2576a^{22}+12397a^{20}+40964a^{18}+94962a^{16}+155040a^{14}+176358a^{12}+136136a^{10}+68068a^{8}+20384a^{6}+3185a^{4}+196a^{2}+2$, $a^{2}+2$, $a^{26}+26a^{24}+299a^{22}+2002a^{20}+8645a^{18}+25194a^{16}+50388a^{14}+68952a^{12}+63206a^{10}+37180a^{8}+13013a^{6}+2366a^{4}+169a^{2}+2$, $a^{35}+35a^{33}+560a^{31}+5425a^{29}+35525a^{27}+166257a^{25}+573300a^{23}+1480050a^{21}+2877875a^{19}+4206125a^{17}+4576264a^{15}+3640210a^{13}+2057510a^{11}+791350a^{9}+193800a^{7}+27132a^{5}+1785a^{3}+35a$, $a^{34}+34a^{32}+527a^{30}+4930a^{28}+31059a^{26}+139230a^{24}+457470a^{22}+1118260a^{20}+2042975a^{18}+2778446a^{16}+2778446a^{14}+1998724a^{12}+999362a^{10}+329460a^{8}+65892a^{6}+6936a^{4}+289a^{2}+2$, $a$, $a^{34}+34a^{32}+527a^{30}+4930a^{28}+31060a^{26}+139256a^{24}+457769a^{22}+1120262a^{20}+2051620a^{18}+2803640a^{16}+2828834a^{14}+2067676a^{12}+1062568a^{10}+366640a^{8}+78905a^{6}+9302a^{4}+458a^{2}+3$, $a^{41}+40a^{39}+741a^{37}+8436a^{35}+66045a^{33}+376992a^{31}+1623160a^{29}+5379617a^{27}+13884183a^{25}+28049124a^{23}+44354442a^{21}+54637695a^{19}+51928254a^{17}+37511928a^{15}+20162952a^{13}+7833567a^{11}+2115916a^{9}+377036a^{7}+41097a^{5}+2414a^{3}+59a$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24753a^{12}+27468a^{10}+19359a^{8}+8120a^{6}+1821a^{4}+180a^{2}+5$, $a^{40}+39a^{38}+703a^{36}+7770a^{34}+58905a^{32}+324632a^{30}+1344903a^{28}+4272021a^{26}+10517975a^{24}+20157774a^{22}+30034368a^{20}+34563451a^{18}+30346173a^{16}+19938960a^{14}+9525542a^{12}+3168375a^{10}+685287a^{8}+85596a^{6}+4816a^{4}+57a^{2}+1$, $a^{41}+40a^{39}+741a^{37}+8436a^{35}+66045a^{33}+376992a^{31}+1623160a^{29}+5379616a^{27}+13884156a^{25}+28048801a^{23}+44352188a^{21}+54627530a^{19}+51897246a^{17}+37446852a^{15}+20069248a^{13}+7742904a^{11}+2059420a^{9}+355971a^{7}+36938a^{5}+2046a^{3}+44a$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273295a^{28}+1076075a^{26}+3223000a^{24}+7411129a^{22}+13110713a^{20}+17768972a^{18}+18254686a^{16}+13960195a^{14}+7728644a^{12}+2970473a^{10}+743798a^{8}+109879a^{6}+8185a^{4}+246a^{2}+2$, $a^{16}+16a^{14}+104a^{12}+352a^{10}+660a^{8}+672a^{6}+336a^{4}+64a^{2}+3$, $a^{14}+14a^{12}+77a^{10}+210a^{8}+294a^{6}+196a^{4}+49a^{2}+2$, $a^{28}+28a^{26}+350a^{24}+2576a^{22}+12397a^{20}+40964a^{18}+94962a^{16}+155039a^{14}+176344a^{12}+136059a^{10}+67858a^{8}+20090a^{6}+2989a^{4}+147a^{2}+1$, $a^{41}+40a^{39}+741a^{37}+8437a^{35}+66080a^{33}+377552a^{31}+1628585a^{29}+5415141a^{27}+14050413a^{25}+28622100a^{23}+45832215a^{21}+57505175a^{19}+56102060a^{17}+42018424a^{15}+23698510a^{13}+9783670a^{11}+2834325a^{9}+539904a^{7}+60781a^{5}+3325a^{3}+56a$, $a^{35}+35a^{33}+560a^{31}+5425a^{29}+35525a^{27}+166257a^{25}+573300a^{23}+1480050a^{21}+2877876a^{19}+4206144a^{17}+4576416a^{15}+3640875a^{13}+2059239a^{11}+794067a^{9}+196308a^{7}+28386a^{5}+2070a^{3}+54a$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 2748021948787771.5 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{21}\cdot 2748021948787771.5 \cdot 3756652}{4\cdot\sqrt{959396304051793463814262846982490027578741814649477038563926538598268329263104}}\cr\approx \mathstrut & 0.152243719791663 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_{42}$ (as 42T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.3.1849.1, 6.0.218803264.1, 7.7.6321363049.1, 14.0.654698590982350051753984.1, \(\Q(\zeta_{43})^+\)

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $42$ $21^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{7}$ ${\href{/padicField/11.14.0.1}{14} }^{3}$ $21^{2}$ $21^{2}$ $42$ $42$ $21^{2}$ $42$ ${\href{/padicField/37.3.0.1}{3} }^{14}$ ${\href{/padicField/41.7.0.1}{7} }^{6}$ R ${\href{/padicField/47.14.0.1}{14} }^{3}$ $21^{2}$ ${\href{/padicField/59.14.0.1}{14} }^{3}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.14.14.38$x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$$2$$7$$14$$C_{14}$$[2]^{7}$
\(43\) Copy content Toggle raw display Deg $42$$21$$2$$40$