Properties

Label 42.0.475...976.1
Degree $42$
Signature $[0, 21]$
Discriminant $-4.751\times 10^{69}$
Root discriminant $45.60$
Ramified primes $2, 29$
Class number not computed
Class group not computed
Galois group $C_7\times S_3$ (as 42T6)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 46*x^40 + 825*x^38 + 9020*x^36 + 65490*x^34 + 325184*x^32 + 1115476*x^30 + 2295041*x^28 + 2315947*x^26 - 2734559*x^24 - 14848773*x^22 - 6221636*x^20 + 35462359*x^18 - 4544065*x^16 - 16428325*x^14 + 5957296*x^12 + 1133059*x^10 - 497379*x^8 - 50982*x^6 + 12238*x^4 + 1537*x^2 + 29)
 
gp: K = bnfinit(x^42 + 46*x^40 + 825*x^38 + 9020*x^36 + 65490*x^34 + 325184*x^32 + 1115476*x^30 + 2295041*x^28 + 2315947*x^26 - 2734559*x^24 - 14848773*x^22 - 6221636*x^20 + 35462359*x^18 - 4544065*x^16 - 16428325*x^14 + 5957296*x^12 + 1133059*x^10 - 497379*x^8 - 50982*x^6 + 12238*x^4 + 1537*x^2 + 29, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 0, 1537, 0, 12238, 0, -50982, 0, -497379, 0, 1133059, 0, 5957296, 0, -16428325, 0, -4544065, 0, 35462359, 0, -6221636, 0, -14848773, 0, -2734559, 0, 2315947, 0, 2295041, 0, 1115476, 0, 325184, 0, 65490, 0, 9020, 0, 825, 0, 46, 0, 1]);
 

\( x^{42} + 46 x^{40} + 825 x^{38} + 9020 x^{36} + 65490 x^{34} + 325184 x^{32} + 1115476 x^{30} + 2295041 x^{28} + 2315947 x^{26} - 2734559 x^{24} - 14848773 x^{22} - 6221636 x^{20} + 35462359 x^{18} - 4544065 x^{16} - 16428325 x^{14} + 5957296 x^{12} + 1133059 x^{10} - 497379 x^{8} - 50982 x^{6} + 12238 x^{4} + 1537 x^{2} + 29 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-47\!\cdots\!976\)\(\medspace = -\,2^{42}\cdot 29^{39}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.60$
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:  $2, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $42$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{4} a^{28} - \frac{1}{2} a^{27} + \frac{1}{4} a^{26} - \frac{1}{2} a^{25} - \frac{1}{2} a^{24} + \frac{1}{4} a^{22} - \frac{1}{2} a^{19} - \frac{1}{4} a^{18} - \frac{1}{2} a^{17} + \frac{1}{4} a^{16} - \frac{1}{4} a^{14} + \frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{29} + \frac{1}{4} a^{27} - \frac{1}{2} a^{25} + \frac{1}{4} a^{23} - \frac{1}{2} a^{22} - \frac{1}{2} a^{20} - \frac{1}{4} a^{19} + \frac{1}{4} a^{17} - \frac{1}{2} a^{16} - \frac{1}{4} a^{15} - \frac{1}{2} a^{14} + \frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{30} - \frac{1}{2} a^{27} + \frac{1}{4} a^{26} - \frac{1}{2} a^{25} - \frac{1}{4} a^{24} - \frac{1}{2} a^{23} - \frac{1}{4} a^{22} - \frac{1}{2} a^{21} - \frac{1}{4} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{31} + \frac{1}{4} a^{27} - \frac{1}{4} a^{25} - \frac{1}{2} a^{24} - \frac{1}{4} a^{23} - \frac{1}{4} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{32} - \frac{1}{2} a^{27} - \frac{1}{2} a^{26} + \frac{1}{4} a^{24} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{4} a^{18} - \frac{1}{2} a^{17} + \frac{1}{4} a^{16} - \frac{1}{2} a^{15} + \frac{1}{4} a^{14} - \frac{1}{2} a^{13} + \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4}$, $\frac{1}{4} a^{33} - \frac{1}{2} a^{27} - \frac{1}{2} a^{26} + \frac{1}{4} a^{25} - \frac{1}{2} a^{23} - \frac{1}{2} a^{21} - \frac{1}{4} a^{19} + \frac{1}{4} a^{17} + \frac{1}{4} a^{15} + \frac{1}{4} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{34} - \frac{1}{2} a^{27} - \frac{1}{4} a^{26} - \frac{1}{2} a^{24} - \frac{1}{4} a^{20} - \frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{35} - \frac{1}{4} a^{27} - \frac{1}{2} a^{26} - \frac{1}{2} a^{25} - \frac{1}{2} a^{22} - \frac{1}{4} a^{21} - \frac{1}{4} a^{19} - \frac{1}{2} a^{18} - \frac{1}{4} a^{17} - \frac{1}{2} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{764} a^{36} - \frac{9}{764} a^{34} + \frac{7}{191} a^{32} + \frac{83}{764} a^{30} + \frac{2}{191} a^{28} - \frac{233}{764} a^{26} + \frac{145}{764} a^{24} - \frac{303}{764} a^{22} - \frac{1}{2} a^{21} + \frac{57}{764} a^{20} - \frac{1}{2} a^{19} + \frac{205}{764} a^{18} - \frac{353}{764} a^{16} - \frac{1}{2} a^{15} + \frac{61}{764} a^{14} - \frac{83}{764} a^{12} - \frac{49}{764} a^{10} - \frac{1}{2} a^{9} - \frac{127}{764} a^{8} + \frac{70}{191} a^{6} - \frac{1}{2} a^{5} + \frac{139}{382} a^{4} - \frac{1}{2} a^{3} - \frac{3}{382} a^{2} - \frac{1}{2} a + \frac{12}{191}$, $\frac{1}{764} a^{37} - \frac{9}{764} a^{35} + \frac{7}{191} a^{33} + \frac{83}{764} a^{31} + \frac{2}{191} a^{29} - \frac{233}{764} a^{27} + \frac{145}{764} a^{25} - \frac{303}{764} a^{23} - \frac{1}{2} a^{22} + \frac{57}{764} a^{21} - \frac{1}{2} a^{20} + \frac{205}{764} a^{19} - \frac{353}{764} a^{17} - \frac{1}{2} a^{16} + \frac{61}{764} a^{15} - \frac{83}{764} a^{13} - \frac{49}{764} a^{11} - \frac{1}{2} a^{10} - \frac{127}{764} a^{9} + \frac{70}{191} a^{7} - \frac{1}{2} a^{6} + \frac{139}{382} a^{5} - \frac{1}{2} a^{4} - \frac{3}{382} a^{3} - \frac{1}{2} a^{2} + \frac{12}{191} a$, $\frac{1}{764} a^{38} - \frac{53}{764} a^{34} - \frac{47}{764} a^{32} - \frac{9}{764} a^{30} + \frac{15}{382} a^{28} - \frac{1}{2} a^{27} - \frac{233}{764} a^{26} - \frac{1}{2} a^{25} + \frac{119}{382} a^{24} - \frac{1}{2} a^{23} - \frac{187}{764} a^{22} - \frac{23}{382} a^{20} + \frac{155}{764} a^{18} - \frac{251}{764} a^{16} - \frac{1}{2} a^{15} - \frac{107}{764} a^{14} - \frac{223}{764} a^{12} + \frac{5}{764} a^{10} - \frac{1}{2} a^{9} + \frac{23}{191} a^{8} + \frac{315}{764} a^{6} + \frac{13}{764} a^{4} - \frac{197}{764} a^{2} + \frac{241}{764}$, $\frac{1}{764} a^{39} - \frac{53}{764} a^{35} - \frac{47}{764} a^{33} - \frac{9}{764} a^{31} + \frac{15}{382} a^{29} - \frac{233}{764} a^{27} + \frac{119}{382} a^{25} - \frac{1}{2} a^{24} - \frac{187}{764} a^{23} - \frac{1}{2} a^{22} - \frac{23}{382} a^{21} + \frac{155}{764} a^{19} - \frac{1}{2} a^{18} - \frac{251}{764} a^{17} - \frac{107}{764} a^{15} - \frac{1}{2} a^{14} - \frac{223}{764} a^{13} - \frac{1}{2} a^{12} + \frac{5}{764} a^{11} + \frac{23}{191} a^{9} - \frac{1}{2} a^{8} + \frac{315}{764} a^{7} - \frac{1}{2} a^{6} + \frac{13}{764} a^{5} - \frac{1}{2} a^{4} - \frac{197}{764} a^{3} - \frac{1}{2} a^{2} + \frac{241}{764} a - \frac{1}{2}$, $\frac{1}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{40} + \frac{986823787769738277860140886045448586832051087318775515498513012051412818594335}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{38} - \frac{664756697211751911375384794905258173334755798652581173106478976784749276147301}{1256344058126372798733895878489275162697890078070954627682994000944702276872464338} a^{36} - \frac{2426985766746874264940945640292340857180890001201151530129196348340119892250076}{36951295827246258786291055249684563608761472884439841990676294145432419908013657} a^{34} + \frac{30060671072112343948687013945615189881660528989314828801559187707808317662731099}{628172029063186399366947939244637581348945039035477313841497000472351138436232169} a^{32} + \frac{152951633074504088290677180907149930302389200452293706748415946703548473542066779}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{30} + \frac{2627305888935340606599377083719417907036492762772054582075871381873058771735682}{36951295827246258786291055249684563608761472884439841990676294145432419908013657} a^{28} + \frac{705920563702401621851691669902998680216933862875720677909605437686192048733027205}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{26} - \frac{878313425689682699990361710318429394690600555878095119879956724296281508131473055}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{24} - \frac{1}{2} a^{23} + \frac{126223848420633606552257435753503968476897913885810822315397726886083860426861855}{628172029063186399366947939244637581348945039035477313841497000472351138436232169} a^{22} + \frac{96235551863003026360108821355772234870226824987131546974043522603948377805626744}{628172029063186399366947939244637581348945039035477313841497000472351138436232169} a^{20} - \frac{490886002758105674947871385514502594930700256337196211176291285015991362115837099}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{18} - \frac{1}{2} a^{17} - \frac{109064069492252529670465295463693511275472336854965700212758251407065563054791377}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{16} + \frac{1070468100365737563405336662979192572708589466724690865399863029737573780055532711}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{14} - \frac{1122642438324422707962536797339085554028581871577483834086277387305978225861417245}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{12} - \frac{1}{2} a^{11} + \frac{118032167418248185098394130310679997863052016333272837879634117416187968791088931}{1256344058126372798733895878489275162697890078070954627682994000944702276872464338} a^{10} - \frac{1}{2} a^{9} + \frac{425398936685904258123015237091309214638126083958612673369101475458223199690075927}{1256344058126372798733895878489275162697890078070954627682994000944702276872464338} a^{8} + \frac{193471267483262490591554467736343778882020920017188726321417585083215399752309603}{1256344058126372798733895878489275162697890078070954627682994000944702276872464338} a^{6} - \frac{1}{2} a^{5} - \frac{669180744620183184838468853723406402345683818121242466871896985261436687975370323}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{4} - \frac{1}{2} a^{3} - \frac{286870070335351276644617248199577296710831935568784632006318894389756021850106262}{628172029063186399366947939244637581348945039035477313841497000472351138436232169} a^{2} - \frac{438830539318556763441174761534173493733866182250651478986609937637346355705147389}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676}$, $\frac{1}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{41} + \frac{986823787769738277860140886045448586832051087318775515498513012051412818594335}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{39} - \frac{664756697211751911375384794905258173334755798652581173106478976784749276147301}{1256344058126372798733895878489275162697890078070954627682994000944702276872464338} a^{37} - \frac{2426985766746874264940945640292340857180890001201151530129196348340119892250076}{36951295827246258786291055249684563608761472884439841990676294145432419908013657} a^{35} + \frac{30060671072112343948687013945615189881660528989314828801559187707808317662731099}{628172029063186399366947939244637581348945039035477313841497000472351138436232169} a^{33} + \frac{152951633074504088290677180907149930302389200452293706748415946703548473542066779}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{31} + \frac{2627305888935340606599377083719417907036492762772054582075871381873058771735682}{36951295827246258786291055249684563608761472884439841990676294145432419908013657} a^{29} + \frac{705920563702401621851691669902998680216933862875720677909605437686192048733027205}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{27} - \frac{878313425689682699990361710318429394690600555878095119879956724296281508131473055}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{25} - \frac{1}{2} a^{24} + \frac{126223848420633606552257435753503968476897913885810822315397726886083860426861855}{628172029063186399366947939244637581348945039035477313841497000472351138436232169} a^{23} + \frac{96235551863003026360108821355772234870226824987131546974043522603948377805626744}{628172029063186399366947939244637581348945039035477313841497000472351138436232169} a^{21} - \frac{490886002758105674947871385514502594930700256337196211176291285015991362115837099}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{19} - \frac{1}{2} a^{18} - \frac{109064069492252529670465295463693511275472336854965700212758251407065563054791377}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{17} + \frac{1070468100365737563405336662979192572708589466724690865399863029737573780055532711}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{15} - \frac{1122642438324422707962536797339085554028581871577483834086277387305978225861417245}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{13} - \frac{1}{2} a^{12} + \frac{118032167418248185098394130310679997863052016333272837879634117416187968791088931}{1256344058126372798733895878489275162697890078070954627682994000944702276872464338} a^{11} - \frac{1}{2} a^{10} + \frac{425398936685904258123015237091309214638126083958612673369101475458223199690075927}{1256344058126372798733895878489275162697890078070954627682994000944702276872464338} a^{9} + \frac{193471267483262490591554467736343778882020920017188726321417585083215399752309603}{1256344058126372798733895878489275162697890078070954627682994000944702276872464338} a^{7} - \frac{1}{2} a^{6} - \frac{669180744620183184838468853723406402345683818121242466871896985261436687975370323}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a^{5} - \frac{1}{2} a^{4} - \frac{286870070335351276644617248199577296710831935568784632006318894389756021850106262}{628172029063186399366947939244637581348945039035477313841497000472351138436232169} a^{3} - \frac{438830539318556763441174761534173493733866182250651478986609937637346355705147389}{2512688116252745597467791756978550325395780156141909255365988001889404553744928676} a$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $20$
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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_7\times S_3$ (as 42T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 42
The 21 conjugacy class representatives for $C_7\times S_3$
Character table for $C_7\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-29}) \), 3.1.116.1 x3, 6.0.1560896.1, 7.7.594823321.1, 14.0.168110140833113738264576.1, 21.7.99995832264130420565259872976896.1 x3

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 $21^{2}$ $21^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{3}$ $21^{2}$ $21^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{21}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{3}$ R $21^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{21}$ $21^{2}$ $21^{2}$ $21^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{21}$

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
$2$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.116.2t1.a.a$1$ $ 2^{2} \cdot 29 $ $x^{2} + 29$ $C_2$ (as 2T1) $1$ $-1$
* 1.29.7t1.a.a$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.a$1$ $ 2^{2} \cdot 29 $ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.116.14t1.a.b$1$ $ 2^{2} \cdot 29 $ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.b$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.29.7t1.a.c$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.29.7t1.a.d$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.c$1$ $ 2^{2} \cdot 29 $ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.e$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.d$1$ $ 2^{2} \cdot 29 $ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.116.14t1.a.e$1$ $ 2^{2} \cdot 29 $ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.116.14t1.a.f$1$ $ 2^{2} \cdot 29 $ $x^{14} + 29 x^{12} + 290 x^{10} + 1247 x^{8} + 2262 x^{6} + 1566 x^{4} + 377 x^{2} + 29$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.f$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
*2 2.116.3t2.a.a$2$ $ 2^{2} \cdot 29 $ $x^{3} - x^{2} - 2$ $S_3$ (as 3T2) $1$ $0$
*2 2.3364.21t6.b.a$2$ $ 2^{2} \cdot 29^{2}$ $x^{42} + 46 x^{40} + 825 x^{38} + 9020 x^{36} + 65490 x^{34} + 325184 x^{32} + 1115476 x^{30} + 2295041 x^{28} + 2315947 x^{26} - 2734559 x^{24} - 14848773 x^{22} - 6221636 x^{20} + 35462359 x^{18} - 4544065 x^{16} - 16428325 x^{14} + 5957296 x^{12} + 1133059 x^{10} - 497379 x^{8} - 50982 x^{6} + 12238 x^{4} + 1537 x^{2} + 29$ $C_7\times S_3$ (as 42T6) $0$ $0$
*2 2.3364.21t6.b.b$2$ $ 2^{2} \cdot 29^{2}$ $x^{42} + 46 x^{40} + 825 x^{38} + 9020 x^{36} + 65490 x^{34} + 325184 x^{32} + 1115476 x^{30} + 2295041 x^{28} + 2315947 x^{26} - 2734559 x^{24} - 14848773 x^{22} - 6221636 x^{20} + 35462359 x^{18} - 4544065 x^{16} - 16428325 x^{14} + 5957296 x^{12} + 1133059 x^{10} - 497379 x^{8} - 50982 x^{6} + 12238 x^{4} + 1537 x^{2} + 29$ $C_7\times S_3$ (as 42T6) $0$ $0$
*2 2.3364.21t6.b.c$2$ $ 2^{2} \cdot 29^{2}$ $x^{42} + 46 x^{40} + 825 x^{38} + 9020 x^{36} + 65490 x^{34} + 325184 x^{32} + 1115476 x^{30} + 2295041 x^{28} + 2315947 x^{26} - 2734559 x^{24} - 14848773 x^{22} - 6221636 x^{20} + 35462359 x^{18} - 4544065 x^{16} - 16428325 x^{14} + 5957296 x^{12} + 1133059 x^{10} - 497379 x^{8} - 50982 x^{6} + 12238 x^{4} + 1537 x^{2} + 29$ $C_7\times S_3$ (as 42T6) $0$ $0$
*2 2.3364.21t6.b.d$2$ $ 2^{2} \cdot 29^{2}$ $x^{42} + 46 x^{40} + 825 x^{38} + 9020 x^{36} + 65490 x^{34} + 325184 x^{32} + 1115476 x^{30} + 2295041 x^{28} + 2315947 x^{26} - 2734559 x^{24} - 14848773 x^{22} - 6221636 x^{20} + 35462359 x^{18} - 4544065 x^{16} - 16428325 x^{14} + 5957296 x^{12} + 1133059 x^{10} - 497379 x^{8} - 50982 x^{6} + 12238 x^{4} + 1537 x^{2} + 29$ $C_7\times S_3$ (as 42T6) $0$ $0$
*2 2.3364.21t6.b.e$2$ $ 2^{2} \cdot 29^{2}$ $x^{42} + 46 x^{40} + 825 x^{38} + 9020 x^{36} + 65490 x^{34} + 325184 x^{32} + 1115476 x^{30} + 2295041 x^{28} + 2315947 x^{26} - 2734559 x^{24} - 14848773 x^{22} - 6221636 x^{20} + 35462359 x^{18} - 4544065 x^{16} - 16428325 x^{14} + 5957296 x^{12} + 1133059 x^{10} - 497379 x^{8} - 50982 x^{6} + 12238 x^{4} + 1537 x^{2} + 29$ $C_7\times S_3$ (as 42T6) $0$ $0$
*2 2.3364.21t6.b.f$2$ $ 2^{2} \cdot 29^{2}$ $x^{42} + 46 x^{40} + 825 x^{38} + 9020 x^{36} + 65490 x^{34} + 325184 x^{32} + 1115476 x^{30} + 2295041 x^{28} + 2315947 x^{26} - 2734559 x^{24} - 14848773 x^{22} - 6221636 x^{20} + 35462359 x^{18} - 4544065 x^{16} - 16428325 x^{14} + 5957296 x^{12} + 1133059 x^{10} - 497379 x^{8} - 50982 x^{6} + 12238 x^{4} + 1537 x^{2} + 29$ $C_7\times S_3$ (as 42T6) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.