Properties

Label 42.42.776...312.1
Degree $42$
Signature $[42, 0]$
Discriminant $7.767\times 10^{86}$
Root discriminant \(117.17\)
Ramified primes $2,3,7$
Class number not computed
Class group not computed
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 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*x + 1)
 
gp: K = bnfinit(y^42 - 105*y^40 + 4977*y^38 - 140819*y^36 - 2*y^35 + 2650095*y^34 + 28*y^33 - 34991649*y^32 + 3416*y^31 + 333300632*y^30 - 147910*y^29 - 2319579309*y^28 + 2698850*y^27 + 11820066804*y^26 - 26976474*y^25 - 43812582639*y^24 + 153194076*y^23 + 116368453593*y^22 - 445815310*y^21 - 216225058245*y^20 + 272316800*y^19 + 271556250546*y^18 + 1986860386*y^17 - 220004406804*y^16 - 5445985720*y^15 + 108239023559*y^14 + 5365080098*y^13 - 29954210552*y^12 - 2176838734*y^11 + 4303639312*y^10 + 340937478*y^9 - 323375283*y^8 - 25471098*y^7 + 12576760*y^6 + 955878*y^5 - 232127*y^4 - 16898*y^3 + 1505*y^2 + 112*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*x + 1)
 

\( x^{42} - 105 x^{40} + 4977 x^{38} - 140819 x^{36} - 2 x^{35} + 2650095 x^{34} + 28 x^{33} - 34991649 x^{32} + 3416 x^{31} + 333300632 x^{30} - 147910 x^{29} - 2319579309 x^{28} + \cdots + 1 \) 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:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(776\!\cdots\!312\) \(\medspace = 2^{42}\cdot 3^{21}\cdot 7^{76}\) 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:  \(117.17\)
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:  $2\cdot 3^{1/2}7^{38/21}\approx 117.16978899149642$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q(\sqrt{3}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(588=2^{2}\cdot 3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{588}(1,·)$, $\chi_{588}(515,·)$, $\chi_{588}(263,·)$, $\chi_{588}(11,·)$, $\chi_{588}(527,·)$, $\chi_{588}(529,·)$, $\chi_{588}(275,·)$, $\chi_{588}(277,·)$, $\chi_{588}(23,·)$, $\chi_{588}(25,·)$, $\chi_{588}(155,·)$, $\chi_{588}(541,·)$, $\chi_{588}(289,·)$, $\chi_{588}(421,·)$, $\chi_{588}(169,·)$, $\chi_{588}(95,·)$, $\chi_{588}(431,·)$, $\chi_{588}(179,·)$, $\chi_{588}(443,·)$, $\chi_{588}(445,·)$, $\chi_{588}(575,·)$, $\chi_{588}(193,·)$, $\chi_{588}(323,·)$, $\chi_{588}(407,·)$, $\chi_{588}(71,·)$, $\chi_{588}(457,·)$, $\chi_{588}(205,·)$, $\chi_{588}(337,·)$, $\chi_{588}(85,·)$, $\chi_{588}(121,·)$, $\chi_{588}(347,·)$, $\chi_{588}(37,·)$, $\chi_{588}(107,·)$, $\chi_{588}(359,·)$, $\chi_{588}(361,·)$, $\chi_{588}(491,·)$, $\chi_{588}(109,·)$, $\chi_{588}(239,·)$, $\chi_{588}(373,·)$, $\chi_{588}(505,·)$, $\chi_{588}(191,·)$, $\chi_{588}(253,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{97}a^{37}+\frac{45}{97}a^{36}+\frac{16}{97}a^{35}+\frac{34}{97}a^{34}+\frac{18}{97}a^{33}-\frac{26}{97}a^{32}+\frac{45}{97}a^{31}-\frac{27}{97}a^{30}-\frac{24}{97}a^{29}+\frac{22}{97}a^{28}-\frac{6}{97}a^{27}-\frac{19}{97}a^{26}+\frac{20}{97}a^{25}+\frac{35}{97}a^{24}-\frac{37}{97}a^{23}+\frac{39}{97}a^{22}+\frac{46}{97}a^{21}-\frac{20}{97}a^{20}+\frac{12}{97}a^{19}-\frac{27}{97}a^{18}+\frac{39}{97}a^{17}+\frac{2}{97}a^{16}+\frac{18}{97}a^{15}-\frac{37}{97}a^{14}+\frac{33}{97}a^{13}-\frac{6}{97}a^{12}+\frac{22}{97}a^{11}+\frac{12}{97}a^{10}-\frac{34}{97}a^{9}-\frac{31}{97}a^{8}-\frac{46}{97}a^{7}+\frac{28}{97}a^{6}-\frac{24}{97}a^{5}-\frac{18}{97}a^{4}-\frac{48}{97}a^{3}+\frac{28}{97}a^{2}+\frac{11}{97}a+\frac{1}{97}$, $\frac{1}{97}a^{38}+\frac{28}{97}a^{36}-\frac{7}{97}a^{35}+\frac{40}{97}a^{34}+\frac{37}{97}a^{33}-\frac{46}{97}a^{32}-\frac{15}{97}a^{31}+\frac{27}{97}a^{30}+\frac{35}{97}a^{29}-\frac{26}{97}a^{28}-\frac{40}{97}a^{27}+\frac{2}{97}a^{26}+\frac{8}{97}a^{25}+\frac{37}{97}a^{24}-\frac{42}{97}a^{23}+\frac{37}{97}a^{22}+\frac{44}{97}a^{21}+\frac{39}{97}a^{20}+\frac{15}{97}a^{19}-\frac{7}{97}a^{18}-\frac{7}{97}a^{17}+\frac{25}{97}a^{16}+\frac{26}{97}a^{15}-\frac{48}{97}a^{14}-\frac{36}{97}a^{13}+\frac{1}{97}a^{12}-\frac{8}{97}a^{11}+\frac{8}{97}a^{10}+\frac{44}{97}a^{9}-\frac{9}{97}a^{8}-\frac{36}{97}a^{7}-\frac{23}{97}a^{6}-\frac{5}{97}a^{5}-\frac{14}{97}a^{4}-\frac{43}{97}a^{3}+\frac{12}{97}a^{2}-\frac{9}{97}a-\frac{45}{97}$, $\frac{1}{97}a^{39}-\frac{6}{97}a^{36}-\frac{20}{97}a^{35}-\frac{42}{97}a^{34}+\frac{32}{97}a^{33}+\frac{34}{97}a^{32}+\frac{28}{97}a^{31}+\frac{15}{97}a^{30}-\frac{33}{97}a^{29}+\frac{23}{97}a^{28}-\frac{24}{97}a^{27}-\frac{42}{97}a^{26}-\frac{38}{97}a^{25}+\frac{45}{97}a^{24}+\frac{6}{97}a^{23}+\frac{19}{97}a^{22}+\frac{12}{97}a^{21}-\frac{7}{97}a^{20}+\frac{45}{97}a^{19}-\frac{27}{97}a^{18}-\frac{30}{97}a^{16}+\frac{30}{97}a^{15}+\frac{30}{97}a^{14}+\frac{47}{97}a^{13}-\frac{34}{97}a^{12}-\frac{26}{97}a^{11}-\frac{1}{97}a^{10}-\frac{27}{97}a^{9}-\frac{41}{97}a^{8}+\frac{4}{97}a^{7}-\frac{13}{97}a^{6}-\frac{21}{97}a^{5}-\frac{24}{97}a^{4}-\frac{2}{97}a^{3}-\frac{17}{97}a^{2}+\frac{35}{97}a-\frac{28}{97}$, $\frac{1}{97}a^{40}-\frac{41}{97}a^{36}-\frac{43}{97}a^{35}+\frac{42}{97}a^{34}+\frac{45}{97}a^{33}-\frac{31}{97}a^{32}-\frac{6}{97}a^{31}-\frac{1}{97}a^{30}-\frac{24}{97}a^{29}+\frac{11}{97}a^{28}+\frac{19}{97}a^{27}+\frac{42}{97}a^{26}-\frac{29}{97}a^{25}+\frac{22}{97}a^{24}-\frac{9}{97}a^{23}-\frac{45}{97}a^{22}-\frac{22}{97}a^{21}+\frac{22}{97}a^{20}+\frac{45}{97}a^{19}+\frac{32}{97}a^{18}+\frac{10}{97}a^{17}+\frac{42}{97}a^{16}+\frac{41}{97}a^{15}+\frac{19}{97}a^{14}-\frac{30}{97}a^{13}+\frac{35}{97}a^{12}+\frac{34}{97}a^{11}+\frac{45}{97}a^{10}+\frac{46}{97}a^{9}+\frac{12}{97}a^{8}+\frac{2}{97}a^{7}-\frac{47}{97}a^{6}+\frac{26}{97}a^{5}-\frac{13}{97}a^{4}-\frac{14}{97}a^{3}+\frac{9}{97}a^{2}+\frac{38}{97}a+\frac{6}{97}$, $\frac{1}{21\!\cdots\!87}a^{41}+\frac{10\!\cdots\!59}{21\!\cdots\!87}a^{40}-\frac{10\!\cdots\!41}{21\!\cdots\!87}a^{39}+\frac{10\!\cdots\!29}{21\!\cdots\!87}a^{38}-\frac{10\!\cdots\!65}{21\!\cdots\!87}a^{37}+\frac{95\!\cdots\!03}{21\!\cdots\!87}a^{36}+\frac{28\!\cdots\!67}{21\!\cdots\!87}a^{35}+\frac{74\!\cdots\!98}{21\!\cdots\!87}a^{34}+\frac{69\!\cdots\!90}{21\!\cdots\!87}a^{33}+\frac{57\!\cdots\!14}{21\!\cdots\!87}a^{32}-\frac{59\!\cdots\!05}{21\!\cdots\!87}a^{31}-\frac{51\!\cdots\!36}{21\!\cdots\!87}a^{30}+\frac{20\!\cdots\!62}{21\!\cdots\!87}a^{29}-\frac{42\!\cdots\!19}{21\!\cdots\!87}a^{28}-\frac{69\!\cdots\!29}{21\!\cdots\!87}a^{27}-\frac{88\!\cdots\!22}{21\!\cdots\!87}a^{26}-\frac{30\!\cdots\!61}{21\!\cdots\!87}a^{25}+\frac{86\!\cdots\!40}{21\!\cdots\!87}a^{24}-\frac{48\!\cdots\!25}{21\!\cdots\!87}a^{23}+\frac{56\!\cdots\!42}{21\!\cdots\!87}a^{22}-\frac{92\!\cdots\!13}{21\!\cdots\!87}a^{21}-\frac{42\!\cdots\!38}{21\!\cdots\!87}a^{20}-\frac{94\!\cdots\!75}{21\!\cdots\!87}a^{19}-\frac{42\!\cdots\!20}{21\!\cdots\!87}a^{18}+\frac{28\!\cdots\!76}{21\!\cdots\!87}a^{17}-\frac{66\!\cdots\!98}{21\!\cdots\!87}a^{16}-\frac{51\!\cdots\!79}{21\!\cdots\!87}a^{15}+\frac{39\!\cdots\!93}{21\!\cdots\!87}a^{14}+\frac{77\!\cdots\!98}{21\!\cdots\!87}a^{13}+\frac{61\!\cdots\!78}{21\!\cdots\!87}a^{12}-\frac{77\!\cdots\!21}{21\!\cdots\!87}a^{11}+\frac{17\!\cdots\!88}{21\!\cdots\!87}a^{10}-\frac{76\!\cdots\!70}{21\!\cdots\!87}a^{9}-\frac{11\!\cdots\!10}{21\!\cdots\!87}a^{8}+\frac{64\!\cdots\!15}{21\!\cdots\!87}a^{7}-\frac{67\!\cdots\!13}{21\!\cdots\!87}a^{6}+\frac{75\!\cdots\!76}{21\!\cdots\!87}a^{5}+\frac{47\!\cdots\!95}{21\!\cdots\!87}a^{4}+\frac{10\!\cdots\!73}{21\!\cdots\!87}a^{3}-\frac{65\!\cdots\!68}{21\!\cdots\!87}a^{2}+\frac{38\!\cdots\!55}{21\!\cdots\!87}a-\frac{28\!\cdots\!63}{21\!\cdots\!87}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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:  $41$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^42 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*x + 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 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*x + 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 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*x + 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 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*x + 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

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{7})^+\), 6.6.4148928.1, 7.7.13841287201.1, 14.14.6864701899232030692065067008.1, \(\Q(\zeta_{49})^+\)

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 R $42$ R $21^{2}$ ${\href{/padicField/13.7.0.1}{7} }^{6}$ $42$ ${\href{/padicField/19.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/padicField/29.14.0.1}{14} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/padicField/41.14.0.1}{14} }^{3}$ ${\href{/padicField/43.14.0.1}{14} }^{3}$ $21^{2}$ $42$ $21^{2}$

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 Deg $42$$2$$21$$42$
\(3\) Copy content Toggle raw display Deg $42$$2$$21$$21$
\(7\) Copy content Toggle raw display Deg $42$$21$$2$$76$