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Properties

Label 26.2.129...328.1
Degree $26$
Signature $[2, 12]$
Discriminant $1.296\times 10^{39}$
Root discriminant \(31.94\)
Ramified primes $2,191$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{26}$ (as 26T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 8*x^24 + 24*x^22 - 16*x^20 + 176*x^18 - 608*x^16 + 320*x^14 + 3712*x^12 + 1280*x^10 - 17408*x^8 + 22528*x^6 + 98304*x^4 + 65536*x^2 - 8192)
 
gp: K = bnfinit(y^26 - 8*y^24 + 24*y^22 - 16*y^20 + 176*y^18 - 608*y^16 + 320*y^14 + 3712*y^12 + 1280*y^10 - 17408*y^8 + 22528*y^6 + 98304*y^4 + 65536*y^2 - 8192, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 8*x^24 + 24*x^22 - 16*x^20 + 176*x^18 - 608*x^16 + 320*x^14 + 3712*x^12 + 1280*x^10 - 17408*x^8 + 22528*x^6 + 98304*x^4 + 65536*x^2 - 8192);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 8*x^24 + 24*x^22 - 16*x^20 + 176*x^18 - 608*x^16 + 320*x^14 + 3712*x^12 + 1280*x^10 - 17408*x^8 + 22528*x^6 + 98304*x^4 + 65536*x^2 - 8192)
 

\( x^{26} - 8 x^{24} + 24 x^{22} - 16 x^{20} + 176 x^{18} - 608 x^{16} + 320 x^{14} + 3712 x^{12} + \cdots - 8192 \) Copy content Toggle raw display

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1295896264146521980742254147120375267328\) \(\medspace = 2^{39}\cdot 191^{12}\) 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:  \(31.94\)
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:  $2^{3/2}191^{1/2}\approx 39.08964057138413$
Ramified primes:   \(2\), \(191\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{7168}a^{20}+\frac{3}{3584}a^{18}+\frac{3}{1792}a^{16}-\frac{1}{896}a^{14}-\frac{3}{448}a^{12}-\frac{3}{112}a^{8}-\frac{1}{56}a^{6}+\frac{1}{7}a^{2}-\frac{2}{7}$, $\frac{1}{7168}a^{21}+\frac{3}{3584}a^{19}+\frac{3}{1792}a^{17}-\frac{1}{896}a^{15}-\frac{3}{448}a^{13}-\frac{3}{112}a^{9}-\frac{1}{56}a^{7}+\frac{1}{7}a^{3}-\frac{2}{7}a$, $\frac{1}{14336}a^{22}+\frac{1}{3584}a^{18}-\frac{3}{1792}a^{16}+\frac{1}{224}a^{12}-\frac{3}{224}a^{10}+\frac{1}{112}a^{8}+\frac{3}{56}a^{6}+\frac{1}{14}a^{4}-\frac{1}{14}a^{2}-\frac{1}{7}$, $\frac{1}{14336}a^{23}+\frac{1}{3584}a^{19}-\frac{3}{1792}a^{17}+\frac{1}{224}a^{13}-\frac{3}{224}a^{11}+\frac{1}{112}a^{9}+\frac{3}{56}a^{7}+\frac{1}{14}a^{5}-\frac{1}{14}a^{3}-\frac{1}{7}a$, $\frac{1}{8845098938368}a^{24}+\frac{144081863}{4422549469184}a^{22}-\frac{3042471}{138204670912}a^{20}-\frac{433108185}{1105637367296}a^{18}-\frac{43879651}{552818683648}a^{16}+\frac{44456947}{17275583864}a^{14}-\frac{8238465}{3735261376}a^{12}-\frac{813048479}{69102335456}a^{10}+\frac{1026038639}{34551167728}a^{8}-\frac{338755635}{8637791932}a^{6}-\frac{449180335}{4318895966}a^{4}+\frac{524958761}{2159447983}a^{2}+\frac{940912370}{2159447983}$, $\frac{1}{8845098938368}a^{25}+\frac{144081863}{4422549469184}a^{23}-\frac{3042471}{138204670912}a^{21}-\frac{433108185}{1105637367296}a^{19}-\frac{43879651}{552818683648}a^{17}+\frac{44456947}{17275583864}a^{15}-\frac{8238465}{3735261376}a^{13}-\frac{813048479}{69102335456}a^{11}+\frac{1026038639}{34551167728}a^{9}-\frac{338755635}{8637791932}a^{7}-\frac{449180335}{4318895966}a^{5}+\frac{524958761}{2159447983}a^{3}+\frac{940912370}{2159447983}a$ 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

Trivial group, which has order $1$ (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:  $13$
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:   $\frac{127977}{1105637367296}a^{24}-\frac{117532367}{4422549469184}a^{22}+\frac{277316563}{1105637367296}a^{20}-\frac{1115227033}{1105637367296}a^{18}+\frac{542027627}{276409341824}a^{16}-\frac{2041853733}{276409341824}a^{14}+\frac{49094305}{1867630688}a^{12}-\frac{177545771}{4318895966}a^{10}-\frac{1344299255}{34551167728}a^{8}+\frac{256972797}{8637791932}a^{6}+\frac{805034441}{4318895966}a^{4}-\frac{3031885253}{4318895966}a^{2}-\frac{1510883293}{2159447983}$, $\frac{6842627}{804099903488}a^{24}-\frac{3331235}{50256243968}a^{22}+\frac{37743059}{201024975872}a^{20}-\frac{10496933}{100512487936}a^{18}+\frac{86903253}{50256243968}a^{16}-\frac{154201335}{25128121984}a^{14}+\frac{39913}{10611538}a^{12}+\frac{158754423}{6282030496}a^{10}+\frac{168352733}{3141015248}a^{8}-\frac{329339663}{1570507624}a^{6}+\frac{47996220}{196313453}a^{4}+\frac{274981143}{392626906}a^{2}+\frac{201170507}{196313453}$, $\frac{78463481}{2211274734592}a^{24}-\frac{389106603}{1105637367296}a^{22}+\frac{1613239619}{1105637367296}a^{20}-\frac{1557155345}{552818683648}a^{18}+\frac{2720125277}{276409341824}a^{16}-\frac{10503738335}{276409341824}a^{14}+\frac{7978615}{116726918}a^{12}+\frac{800815865}{17275583864}a^{10}-\frac{3128914491}{34551167728}a^{8}-\frac{9544337675}{17275583864}a^{6}+\frac{14959570679}{8637791932}a^{4}+\frac{3953016337}{4318895966}a^{2}-\frac{3040971023}{2159447983}$, $\frac{243799209}{8845098938368}a^{24}-\frac{268764169}{1105637367296}a^{22}+\frac{954655351}{1105637367296}a^{20}-\frac{41091877}{34551167728}a^{18}+\frac{3362658553}{552818683648}a^{16}-\frac{1530733735}{69102335456}a^{14}+\frac{100137039}{3735261376}a^{12}+\frac{2673182207}{34551167728}a^{10}-\frac{320442281}{34551167728}a^{8}-\frac{7047144733}{17275583864}a^{6}+\frac{8444954583}{8637791932}a^{4}+\frac{8680601757}{4318895966}a^{2}+\frac{1133579864}{2159447983}$, $\frac{155762235}{8845098938368}a^{24}-\frac{558634435}{4422549469184}a^{22}+\frac{73930867}{276409341824}a^{20}+\frac{244547761}{552818683648}a^{18}+\frac{359766955}{276409341824}a^{16}-\frac{1375437125}{276409341824}a^{14}-\frac{13613305}{933815344}a^{12}+\frac{7648224043}{69102335456}a^{10}+\frac{11282961}{4318895966}a^{8}-\frac{6153614745}{17275583864}a^{6}+\frac{478429568}{2159447983}a^{4}+\frac{5401424506}{2159447983}a^{2}+\frac{3188383213}{2159447983}$, $\frac{2277941}{69102335456}a^{24}-\frac{573981553}{2211274734592}a^{22}+\frac{842980681}{1105637367296}a^{20}-\frac{486717663}{1105637367296}a^{18}+\frac{1518569461}{276409341824}a^{16}-\frac{1224213981}{69102335456}a^{14}+\frac{18726595}{3735261376}a^{12}+\frac{9081147121}{69102335456}a^{10}-\frac{7984061}{4318895966}a^{8}-\frac{8049711975}{17275583864}a^{6}+\frac{5350242729}{8637791932}a^{4}+\frac{6723014161}{2159447983}a^{2}+\frac{4845707445}{2159447983}$, $\frac{4591947}{215734120448}a^{25}-\frac{14338605}{107867060224}a^{23}+\frac{9160779}{53933530112}a^{21}+\frac{12384741}{13483382528}a^{19}+\frac{6456853}{3370845632}a^{17}-\frac{16514853}{3370845632}a^{15}-\frac{1125459}{45551968}a^{13}+\frac{52748001}{421355704}a^{11}+\frac{12825031}{105338926}a^{9}-\frac{179847163}{421355704}a^{7}-\frac{49189047}{210677852}a^{5}+\frac{203809747}{52669463}a^{3}+\frac{205652969}{52669463}a-1$, $\frac{11545365}{215734120448}a^{25}+\frac{97038973}{8845098938368}a^{24}-\frac{23718325}{53933530112}a^{23}-\frac{344418859}{4422549469184}a^{22}+\frac{74988105}{53933530112}a^{21}+\frac{335272957}{2211274734592}a^{20}-\frac{33575669}{26966765056}a^{19}+\frac{108628071}{276409341824}a^{18}+\frac{135670455}{13483382528}a^{17}+\frac{283349257}{552818683648}a^{16}-\frac{60741595}{1685422816}a^{15}-\frac{1073131811}{276409341824}a^{14}+\frac{2802475}{91103936}a^{13}-\frac{10061791}{3735261376}a^{12}+\frac{156076857}{842711408}a^{11}+\frac{4053286635}{69102335456}a^{10}+\frac{2110109}{210677852}a^{9}+\frac{9353017}{4318895966}a^{8}-\frac{385900135}{421355704}a^{7}-\frac{2742273729}{17275583864}a^{6}+\frac{344600589}{210677852}a^{5}+\frac{204104471}{2159447983}a^{4}+\frac{234371006}{52669463}a^{3}+\frac{4591942147}{4318895966}a^{2}+\frac{132088993}{52669463}a+\frac{3425324613}{2159447983}$, $\frac{751147401}{8845098938368}a^{25}-\frac{796753563}{8845098938368}a^{24}-\frac{398962871}{552818683648}a^{23}+\frac{1846521071}{2211274734592}a^{22}+\frac{5272481517}{2211274734592}a^{21}-\frac{446460099}{138204670912}a^{20}-\frac{336709243}{138204670912}a^{19}+\frac{1557267515}{276409341824}a^{18}+\frac{8669495343}{552818683648}a^{17}-\frac{6432600173}{276409341824}a^{16}-\frac{498871843}{8637791932}a^{15}+\frac{5831272287}{69102335456}a^{14}+\frac{180613109}{3735261376}a^{13}-\frac{254418261}{1867630688}a^{12}+\frac{21355926775}{69102335456}a^{11}-\frac{10357800881}{69102335456}a^{10}-\frac{1428121521}{17275583864}a^{9}+\frac{1653883589}{17275583864}a^{8}-\frac{11436679125}{8637791932}a^{7}+\frac{23403010275}{17275583864}a^{6}+\frac{10618636615}{4318895966}a^{5}-\frac{7584321454}{2159447983}a^{4}+\frac{31939833231}{4318895966}a^{3}-\frac{8076590024}{2159447983}a^{2}+\frac{3127564646}{2159447983}a+\frac{709156424}{2159447983}$, $\frac{65012707}{232765761536}a^{25}+\frac{493165521}{8845098938368}a^{24}-\frac{141576137}{58191440384}a^{23}-\frac{704832897}{1105637367296}a^{22}+\frac{985950067}{116382880768}a^{21}+\frac{1711918217}{552818683648}a^{20}-\frac{634725517}{58191440384}a^{19}-\frac{8499989979}{1105637367296}a^{18}+\frac{1704710465}{29095720192}a^{17}+\frac{12050165497}{552818683648}a^{16}-\frac{387935157}{1818482512}a^{15}-\frac{11468444051}{138204670912}a^{14}+\frac{24035821}{98296352}a^{13}+\frac{174906037}{933815344}a^{12}+\frac{3020188565}{3636965024}a^{11}-\frac{1132888301}{17275583864}a^{10}-\frac{351408411}{1818482512}a^{9}-\frac{4391576641}{17275583864}a^{8}-\frac{3995790087}{909241256}a^{7}-\frac{9967535449}{17275583864}a^{6}+\frac{1022296925}{113655157}a^{5}+\frac{32345990725}{8637791932}a^{4}+\frac{2316140678}{113655157}a^{3}-\frac{11175207429}{4318895966}a^{2}+\frac{672117571}{113655157}a-\frac{4138691916}{2159447983}$, $\frac{475918159}{8845098938368}a^{25}-\frac{90995145}{8845098938368}a^{24}-\frac{2285341549}{4422549469184}a^{23}+\frac{305373741}{4422549469184}a^{22}+\frac{4891818517}{2211274734592}a^{21}+\frac{136033949}{2211274734592}a^{20}-\frac{1410056243}{276409341824}a^{19}-\frac{1321379435}{552818683648}a^{18}+\frac{2643001085}{138204670912}a^{17}+\frac{635935935}{69102335456}a^{16}-\frac{8313420729}{138204670912}a^{15}-\frac{6691749425}{276409341824}a^{14}+\frac{418653139}{3735261376}a^{13}+\frac{168845697}{1867630688}a^{12}+\frac{604596695}{34551167728}a^{11}-\frac{23271684393}{69102335456}a^{10}-\frac{3712698365}{34551167728}a^{9}+\frac{9856161727}{17275583864}a^{8}-\frac{468616529}{8637791932}a^{7}-\frac{1081039089}{8637791932}a^{6}+\frac{1583417518}{2159447983}a^{5}-\frac{10051381965}{8637791932}a^{4}+\frac{10574459345}{4318895966}a^{3}-\frac{1212323877}{4318895966}a^{2}+\frac{6005528456}{2159447983}a+\frac{2741981959}{2159447983}$, $\frac{4551090541}{8845098938368}a^{25}+\frac{848443413}{4422549469184}a^{24}-\frac{17944023873}{4422549469184}a^{23}-\frac{852761655}{552818683648}a^{22}+\frac{3284982367}{276409341824}a^{21}+\frac{5218606641}{1105637367296}a^{20}-\frac{3817586393}{552818683648}a^{19}-\frac{4172609121}{1105637367296}a^{18}+\frac{49837278167}{552818683648}a^{17}+\frac{9893524127}{276409341824}a^{16}-\frac{42051247373}{138204670912}a^{15}-\frac{16620701089}{138204670912}a^{14}+\frac{248976657}{1867630688}a^{13}+\frac{140116169}{1867630688}a^{12}+\frac{66310855167}{34551167728}a^{11}+\frac{22979222537}{34551167728}a^{10}+\frac{31429075725}{34551167728}a^{9}+\frac{9148370883}{34551167728}a^{8}-\frac{76936862657}{8637791932}a^{7}-\frac{54600074035}{17275583864}a^{6}+\frac{45151905301}{4318895966}a^{5}+\frac{9049128587}{2159447983}a^{4}+\frac{225129177005}{4318895966}a^{3}+\frac{38524814381}{2159447983}a^{2}+\frac{86224495087}{2159447983}a+\frac{28105232864}{2159447983}$, $\frac{48824313}{180512223232}a^{25}-\frac{6644901}{63633805312}a^{24}-\frac{1363667825}{631792781312}a^{23}+\frac{7108701}{7954225664}a^{22}+\frac{2040334995}{315896390656}a^{21}-\frac{49114333}{15908451328}a^{20}-\frac{666818157}{157948195328}a^{19}+\frac{1005437}{248569552}a^{18}+\frac{3747386877}{78974097664}a^{17}-\frac{22877179}{994278208}a^{16}-\frac{6422739161}{39487048832}a^{15}+\frac{80285767}{994278208}a^{14}+\frac{10741639}{133402192}a^{13}-\frac{165465}{1679524}a^{12}+\frac{10191349615}{9871762208}a^{11}-\frac{149870337}{497139104}a^{10}+\frac{843423865}{2467940552}a^{9}-\frac{3269041}{124284776}a^{8}-\frac{5662271761}{1233970276}a^{7}+\frac{106123063}{62142388}a^{6}+\frac{6950394657}{1233970276}a^{5}-\frac{164781769}{62142388}a^{4}+\frac{1191301048}{44070367}a^{3}-\frac{145420883}{15535597}a^{2}+\frac{6558054654}{308492569}a-\frac{106227845}{15535597}$ 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:  \( 1293801547.1375248 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 1293801547.1375248 \cdot 1}{2\cdot\sqrt{1295896264146521980742254147120375267328}}\cr\approx \mathstrut & 0.272126579712830 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^26 - 8*x^24 + 24*x^22 - 16*x^20 + 176*x^18 - 608*x^16 + 320*x^14 + 3712*x^12 + 1280*x^10 - 17408*x^8 + 22528*x^6 + 98304*x^4 + 65536*x^2 - 8192)
 
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^26 - 8*x^24 + 24*x^22 - 16*x^20 + 176*x^18 - 608*x^16 + 320*x^14 + 3712*x^12 + 1280*x^10 - 17408*x^8 + 22528*x^6 + 98304*x^4 + 65536*x^2 - 8192, 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^26 - 8*x^24 + 24*x^22 - 16*x^20 + 176*x^18 - 608*x^16 + 320*x^14 + 3712*x^12 + 1280*x^10 - 17408*x^8 + 22528*x^6 + 98304*x^4 + 65536*x^2 - 8192);
 
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^26 - 8*x^24 + 24*x^22 - 16*x^20 + 176*x^18 - 608*x^16 + 320*x^14 + 3712*x^12 + 1280*x^10 - 17408*x^8 + 22528*x^6 + 98304*x^4 + 65536*x^2 - 8192);
 
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

$D_{26}$ (as 26T3):

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 solvable group of order 52
The 16 conjugacy class representatives for $D_{26}$
Character table for $D_{26}$

Intermediate fields

\(\Q(\sqrt{2}) \), 13.1.48551226272641.1

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]
 

Sibling fields

Degree 26 sibling: deg 26
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $26$ $26$ ${\href{/padicField/7.2.0.1}{2} }^{12}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{13}$ $26$ ${\href{/padicField/17.13.0.1}{13} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{13}$ ${\href{/padicField/23.13.0.1}{13} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{13}$ ${\href{/padicField/31.2.0.1}{2} }^{12}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{13}$ ${\href{/padicField/41.2.0.1}{2} }^{12}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $26$ ${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{13}$ $26$

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 $26$$2$$13$$39$
\(191\) Copy content Toggle raw display $\Q_{191}$$x + 172$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 172$$1$$1$$0$Trivial$[\ ]$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1528.2t1.b.a$1$ $ 2^{3} \cdot 191 $ \(\Q(\sqrt{-382}) \) $C_2$ (as 2T1) $1$ $-1$
1.191.2t1.a.a$1$ $ 191 $ \(\Q(\sqrt{-191}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
* 2.12224.26t3.a.e$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.e$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.12224.26t3.a.d$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.12224.26t3.a.b$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.d$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.c$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.a$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.12224.26t3.a.c$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.12224.26t3.a.f$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.12224.26t3.a.a$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (as 26T3) $1$ $0$

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 *.