Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 12*x^24 + 44*x^22 - 24*x^20 - 272*x^18 + 1312*x^16 - 1408*x^14 - 3328*x^12 + 30208*x^10 - 19456*x^8 + 46080*x^6 + 204800*x^4 + 131072*x^2 - 8192)
gp: K = bnfinit(y^26 - 12*y^24 + 44*y^22 - 24*y^20 - 272*y^18 + 1312*y^16 - 1408*y^14 - 3328*y^12 + 30208*y^10 - 19456*y^8 + 46080*y^6 + 204800*y^4 + 131072*y^2 - 8192, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 12*x^24 + 44*x^22 - 24*x^20 - 272*x^18 + 1312*x^16 - 1408*x^14 - 3328*x^12 + 30208*x^10 - 19456*x^8 + 46080*x^6 + 204800*x^4 + 131072*x^2 - 8192);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 12*x^24 + 44*x^22 - 24*x^20 - 272*x^18 + 1312*x^16 - 1408*x^14 - 3328*x^12 + 30208*x^10 - 19456*x^8 + 46080*x^6 + 204800*x^4 + 131072*x^2 - 8192)
\( x^{26} - 12 x^{24} + 44 x^{22} - 24 x^{20} - 272 x^{18} + 1312 x^{16} - 1408 x^{14} - 3328 x^{12} + \cdots - 8192 \)
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: |
\(60205866708934265598847651067237790384128\)
\(\medspace = 2^{39}\cdot 263^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(37.02\) | 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: | $2^{3/2}263^{1/2}\approx 45.86937976471886$ | ||
| Ramified primes: |
\(2\), \(263\)
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{640}a^{14}+\frac{1}{160}a^{12}+\frac{1}{80}a^{8}+\frac{1}{20}a^{6}-\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{640}a^{15}+\frac{1}{160}a^{13}+\frac{1}{80}a^{9}+\frac{1}{20}a^{7}-\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{1280}a^{16}+\frac{1}{320}a^{12}+\frac{1}{160}a^{10}+\frac{1}{40}a^{6}-\frac{1}{10}a^{4}-\frac{2}{5}$, $\frac{1}{1280}a^{17}+\frac{1}{320}a^{13}+\frac{1}{160}a^{11}+\frac{1}{40}a^{7}-\frac{1}{10}a^{5}-\frac{2}{5}a$, $\frac{1}{2560}a^{18}-\frac{1}{320}a^{12}+\frac{1}{40}a^{6}-\frac{1}{5}$, $\frac{1}{2560}a^{19}-\frac{1}{320}a^{13}+\frac{1}{40}a^{7}-\frac{1}{5}a$, $\frac{1}{5120}a^{20}+\frac{1}{160}a^{12}+\frac{1}{40}a^{8}+\frac{1}{20}a^{6}+\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5120}a^{21}+\frac{1}{160}a^{13}+\frac{1}{40}a^{9}+\frac{1}{20}a^{7}+\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{51200}a^{22}-\frac{1}{12800}a^{20}-\frac{1}{12800}a^{18}+\frac{1}{6400}a^{16}+\frac{1}{3200}a^{14}-\frac{1}{400}a^{12}-\frac{7}{800}a^{10}+\frac{1}{200}a^{8}+\frac{9}{200}a^{6}-\frac{3}{25}a^{2}-\frac{9}{25}$, $\frac{1}{51200}a^{23}-\frac{1}{12800}a^{21}-\frac{1}{12800}a^{19}+\frac{1}{6400}a^{17}+\frac{1}{3200}a^{15}-\frac{1}{400}a^{13}-\frac{7}{800}a^{11}+\frac{1}{200}a^{9}+\frac{9}{200}a^{7}-\frac{3}{25}a^{3}-\frac{9}{25}a$, $\frac{1}{4123689472000}a^{24}-\frac{15094789}{2061844736000}a^{22}-\frac{4680527}{1030922368000}a^{20}-\frac{8406689}{64432648000}a^{18}-\frac{18518299}{64432648000}a^{16}-\frac{24724933}{64432648000}a^{14}-\frac{79215211}{16108162000}a^{12}-\frac{321924899}{32216324000}a^{10}-\frac{5408989}{460233200}a^{8}-\frac{5475909}{4027040500}a^{6}+\frac{134919957}{2013520250}a^{4}+\frac{67403769}{1006760125}a^{2}+\frac{214299153}{1006760125}$, $\frac{1}{4123689472000}a^{25}-\frac{15094789}{2061844736000}a^{23}-\frac{4680527}{1030922368000}a^{21}-\frac{8406689}{64432648000}a^{19}-\frac{18518299}{64432648000}a^{17}-\frac{24724933}{64432648000}a^{15}-\frac{79215211}{16108162000}a^{13}-\frac{321924899}{32216324000}a^{11}-\frac{5408989}{460233200}a^{9}-\frac{5475909}{4027040500}a^{7}+\frac{134919957}{2013520250}a^{5}+\frac{67403769}{1006760125}a^{3}+\frac{214299153}{1006760125}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
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 $2$)
| 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: |
$\frac{822539}{100577792000}a^{24}-\frac{5137501}{50288896000}a^{22}+\frac{2587433}{6286112000}a^{20}-\frac{5228463}{12572224000}a^{18}-\frac{11919149}{6286112000}a^{16}+\frac{8728499}{785764000}a^{14}-\frac{27279521}{1571528000}a^{12}-\frac{4130613}{392882000}a^{10}+\frac{2718913}{11225200}a^{8}-\frac{52396447}{196441000}a^{6}+\frac{63413571}{98220500}a^{4}+\frac{30188756}{24555125}a^{2}+\frac{27777637}{24555125}$, $\frac{9191153}{515461184000}a^{24}-\frac{468547671}{2061844736000}a^{22}+\frac{960495247}{1030922368000}a^{20}-\frac{457427253}{515461184000}a^{18}-\frac{1218710289}{257730592000}a^{16}+\frac{103278533}{4027040500}a^{14}-\frac{2460373091}{64432648000}a^{12}-\frac{688172953}{16108162000}a^{10}+\frac{127095751}{230116600}a^{8}-\frac{1351061293}{2013520250}a^{6}+\frac{3895632761}{4027040500}a^{4}+\frac{2485405511}{1006760125}a^{2}+\frac{826887987}{1006760125}$, $\frac{1060163}{206184473600}a^{24}-\frac{430873}{8247378944}a^{22}+\frac{145307}{1288652960}a^{20}+\frac{29296003}{103092236800}a^{18}-\frac{1259331}{805408100}a^{16}+\frac{108802871}{25773059200}a^{14}+\frac{5567661}{1288652960}a^{12}-\frac{12174529}{402704050}a^{10}+\frac{28147449}{230116600}a^{8}+\frac{75105509}{402704050}a^{6}+\frac{41782443}{805408100}a^{4}+\frac{491306243}{402704050}a^{2}+\frac{383370394}{201352025}$, $\frac{20540091}{4123689472000}a^{24}-\frac{128866809}{2061844736000}a^{22}+\frac{15949393}{64432648000}a^{20}-\frac{90870107}{515461184000}a^{18}-\frac{201899473}{128865296000}a^{16}+\frac{241162021}{32216324000}a^{14}-\frac{620724589}{64432648000}a^{12}-\frac{30608779}{2013520250}a^{10}+\frac{72560239}{460233200}a^{8}-\frac{976816653}{8054081000}a^{6}+\frac{341683249}{4027040500}a^{4}+\frac{2575463293}{2013520250}a^{2}+\frac{627861238}{1006760125}$, $\frac{17510887}{2061844736000}a^{24}-\frac{211905041}{2061844736000}a^{22}+\frac{101324603}{257730592000}a^{20}-\frac{199243683}{515461184000}a^{18}-\frac{382793709}{257730592000}a^{16}+\frac{314716909}{32216324000}a^{14}-\frac{933129361}{64432648000}a^{12}-\frac{326263491}{32216324000}a^{10}+\frac{24687217}{115058300}a^{8}-\frac{403998213}{2013520250}a^{6}+\frac{712749934}{1006760125}a^{4}+\frac{1590455817}{2013520250}a^{2}+\frac{1691596542}{1006760125}$, $\frac{64898321}{4123689472000}a^{24}-\frac{207513527}{1030922368000}a^{22}+\frac{439122689}{515461184000}a^{20}-\frac{271688021}{257730592000}a^{18}-\frac{225054119}{64432648000}a^{16}+\frac{3072466629}{128865296000}a^{14}-\frac{2671184959}{64432648000}a^{12}-\frac{348486367}{16108162000}a^{10}+\frac{57079091}{115058300}a^{8}-\frac{754193721}{1006760125}a^{6}+\frac{5339888069}{4027040500}a^{4}+\frac{4648254433}{2013520250}a^{2}+\frac{157009703}{1006760125}$, $\frac{2399973}{100577792000}a^{25}-\frac{1671879}{6286112000}a^{23}+\frac{19960399}{25144448000}a^{21}+\frac{5910659}{12572224000}a^{19}-\frac{24046009}{3143056000}a^{17}+\frac{85289847}{3143056000}a^{15}-\frac{8701747}{1571528000}a^{13}-\frac{102803607}{785764000}a^{11}+\frac{7909521}{11225200}a^{9}+\frac{12551473}{98220500}a^{7}+\frac{14601511}{49110250}a^{5}+\frac{325287259}{49110250}a^{3}+\frac{135720259}{24555125}a+1$, $\frac{49675529}{2061844736000}a^{25}-\frac{3347973}{4123689472000}a^{24}-\frac{609518377}{2061844736000}a^{23}+\frac{22116201}{1030922368000}a^{22}+\frac{1191999589}{1030922368000}a^{21}-\frac{225216689}{1030922368000}a^{20}-\frac{143959839}{128865296000}a^{19}+\frac{597400471}{515461184000}a^{18}-\frac{661317679}{128865296000}a^{17}-\frac{926497237}{257730592000}a^{16}+\frac{996971883}{32216324000}a^{15}+\frac{791792473}{128865296000}a^{14}-\frac{2959405867}{64432648000}a^{13}-\frac{77359183}{64432648000}a^{12}-\frac{896548587}{32216324000}a^{11}-\frac{524672983}{32216324000}a^{10}+\frac{3463564}{5752915}a^{9}+\frac{8144039}{230116600}a^{8}-\frac{2873960529}{8054081000}a^{7}+\frac{549646059}{8054081000}a^{6}+\frac{2255400631}{2013520250}a^{5}-\frac{1152281647}{4027040500}a^{4}+\frac{9071066119}{2013520250}a^{3}+\frac{370609448}{1006760125}a^{2}+\frac{2729733359}{1006760125}a+\frac{1073296936}{1006760125}$, $\frac{808546321}{4123689472000}a^{25}-\frac{126879569}{1030922368000}a^{24}-\frac{5136173419}{2061844736000}a^{23}+\frac{3141860089}{2061844736000}a^{22}+\frac{673359913}{64432648000}a^{21}-\frac{6179263123}{1030922368000}a^{20}-\frac{6641102077}{515461184000}a^{19}+\frac{1317923031}{257730592000}a^{18}-\frac{10636947491}{257730592000}a^{17}+\frac{4242488673}{128865296000}a^{16}+\frac{36661383989}{128865296000}a^{15}-\frac{1444714749}{8054081000}a^{14}-\frac{15888325837}{32216324000}a^{13}+\frac{7858241997}{32216324000}a^{12}-\frac{7022821579}{32216324000}a^{11}+\frac{5706737637}{16108162000}a^{10}+\frac{2744047121}{460233200}a^{9}-\frac{114858911}{28764575}a^{8}-\frac{65887496653}{8054081000}a^{7}+\frac{33266338193}{8054081000}a^{6}+\frac{33793489997}{2013520250}a^{5}-\frac{28122920489}{4027040500}a^{4}+\frac{53700421673}{2013520250}a^{3}-\frac{25509793569}{1006760125}a^{2}+\frac{12204976038}{1006760125}a-\frac{2233063828}{1006760125}$, $\frac{23487069}{412368947200}a^{25}-\frac{15643161}{2061844736000}a^{24}-\frac{17223281}{25773059200}a^{23}+\frac{31886977}{515461184000}a^{22}+\frac{472048397}{206184473600}a^{21}-\frac{1740489}{257730592000}a^{20}-\frac{21494197}{103092236800}a^{19}-\frac{169976743}{257730592000}a^{18}-\frac{932575827}{51546118400}a^{17}+\frac{17669537}{257730592000}a^{16}+\frac{461739907}{6443264800}a^{15}+\frac{239635377}{128865296000}a^{14}-\frac{509231911}{12886529600}a^{13}-\frac{508479857}{64432648000}a^{12}-\frac{435474351}{1610816200}a^{11}-\frac{707515647}{32216324000}a^{10}+\frac{384682653}{230116600}a^{9}-\frac{36669557}{460233200}a^{8}-\frac{122638063}{322163240}a^{7}-\frac{1254085429}{8054081000}a^{6}+\frac{515064281}{402704050}a^{5}-\frac{2880927333}{4027040500}a^{4}+\frac{2565752909}{201352025}a^{3}-\frac{1294019793}{1006760125}a^{2}+\frac{406354789}{40270405}a-\frac{2106329966}{1006760125}$, $\frac{79047173}{1030922368000}a^{25}-\frac{13192523}{257730592000}a^{24}-\frac{1876245633}{2061844736000}a^{23}+\frac{1333612937}{2061844736000}a^{22}+\frac{3364815981}{1030922368000}a^{21}-\frac{688378671}{257730592000}a^{20}-\frac{395968617}{257730592000}a^{19}+\frac{773506003}{257730592000}a^{18}-\frac{5281297827}{257730592000}a^{17}+\frac{3058536063}{257730592000}a^{16}+\frac{1549580651}{16108162000}a^{15}-\frac{1219035719}{16108162000}a^{14}-\frac{5819665193}{64432648000}a^{13}+\frac{8137218227}{64432648000}a^{12}-\frac{8676361143}{32216324000}a^{11}+\frac{580448303}{8054081000}a^{10}+\frac{1032845543}{460233200}a^{9}-\frac{718277641}{460233200}a^{8}-\frac{8252556861}{8054081000}a^{7}+\frac{15457738239}{8054081000}a^{6}+\frac{6575196919}{2013520250}a^{5}-\frac{7463107401}{2013520250}a^{4}+\frac{16286252583}{1006760125}a^{3}-\frac{15087281219}{2013520250}a^{2}+\frac{14264387856}{1006760125}a-\frac{2893365919}{1006760125}$, $\frac{197717817}{4123689472000}a^{25}-\frac{327247}{32216324000}a^{24}-\frac{1199243593}{2061844736000}a^{23}+\frac{264169769}{2061844736000}a^{22}+\frac{1143180363}{515461184000}a^{21}-\frac{469900083}{1030922368000}a^{20}-\frac{221054431}{128865296000}a^{19}-\frac{38188939}{257730592000}a^{18}-\frac{1624442881}{128865296000}a^{17}+\frac{1023087031}{257730592000}a^{16}+\frac{2151106437}{32216324000}a^{15}-\frac{1525989699}{128865296000}a^{14}-\frac{1282863807}{16108162000}a^{13}+\frac{91403703}{8054081000}a^{12}-\frac{4353831523}{32216324000}a^{11}+\frac{239167343}{4027040500}a^{10}+\frac{684999671}{460233200}a^{9}-\frac{143706397}{460233200}a^{8}-\frac{9694818401}{8054081000}a^{7}+\frac{230701796}{1006760125}a^{6}+\frac{5868247019}{2013520250}a^{5}+\frac{1804189001}{4027040500}a^{4}+\frac{20819337701}{2013520250}a^{3}-\frac{3039905153}{2013520250}a^{2}+\frac{5385337721}{1006760125}a-\frac{1009934028}{1006760125}$, $\frac{2494067193}{2061844736000}a^{25}+\frac{56064877}{217036288000}a^{24}-\frac{14986822927}{1030922368000}a^{23}-\frac{336724783}{108518144000}a^{22}+\frac{55214576453}{1030922368000}a^{21}+\frac{310785203}{27129536000}a^{20}-\frac{7970764341}{257730592000}a^{19}-\frac{190795819}{27129536000}a^{18}-\frac{83641259881}{257730592000}a^{17}-\frac{454563911}{6782384000}a^{16}+\frac{204635441099}{128865296000}a^{15}+\frac{1135155719}{3391192000}a^{14}-\frac{112650841509}{64432648000}a^{13}-\frac{318383217}{847798000}a^{12}-\frac{125012424839}{32216324000}a^{11}-\frac{662767169}{847798000}a^{10}+\frac{1047512806}{28764575}a^{9}+\frac{23196357}{3027850}a^{8}-\frac{195804544073}{8054081000}a^{7}-\frac{1082260303}{211949500}a^{6}+\frac{236170352779}{4027040500}a^{5}+\frac{2833050403}{211949500}a^{4}+\frac{489963383293}{2013520250}a^{3}+\frac{5304449281}{105974750}a^{2}+\frac{161505938033}{1006760125}a+\frac{2059939901}{52987375}$
| 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: | \( 12785934187.300812 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 12785934187.300812 \cdot 1}{2\cdot\sqrt{60205866708934265598847651067237790384128}}\cr\approx \mathstrut & 0.394549479009867 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 12*x^24 + 44*x^22 - 24*x^20 - 272*x^18 + 1312*x^16 - 1408*x^14 - 3328*x^12 + 30208*x^10 - 19456*x^8 + 46080*x^6 + 204800*x^4 + 131072*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 - 12*x^24 + 44*x^22 - 24*x^20 - 272*x^18 + 1312*x^16 - 1408*x^14 - 3328*x^12 + 30208*x^10 - 19456*x^8 + 46080*x^6 + 204800*x^4 + 131072*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 - 12*x^24 + 44*x^22 - 24*x^20 - 272*x^18 + 1312*x^16 - 1408*x^14 - 3328*x^12 + 30208*x^10 - 19456*x^8 + 46080*x^6 + 204800*x^4 + 131072*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 - 12*x^24 + 44*x^22 - 24*x^20 - 272*x^18 + 1312*x^16 - 1408*x^14 - 3328*x^12 + 30208*x^10 - 19456*x^8 + 46080*x^6 + 204800*x^4 + 131072*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
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
| \(\Q(\sqrt{2}) \), 13.1.330928743953809.1 |
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 | $26$ | ${\href{/padicField/5.2.0.1}{2} }^{13}$ | ${\href{/padicField/7.2.0.1}{2} }^{12}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $26$ | $26$ | ${\href{/padicField/17.13.0.1}{13} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{13}$ | ${\href{/padicField/23.13.0.1}{13} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{13}$ | ${\href{/padicField/31.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/41.2.0.1}{2} }^{12}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $26$ | ${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{13}$ | ${\href{/padicField/59.2.0.1}{2} }^{13}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $26$ | $2$ | $13$ | $39$ | |||
|
\(263\)
| $\Q_{263}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{263}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
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 *.