LMFDB - Number field 28.2.295815184798509371659078776791937755272938949172963.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563)

gp: K = bnfinit(y^28 - 14*y^27 + 80*y^26 - 221*y^25 + 397*y^24 - 1904*y^23 + 10213*y^22 - 26576*y^21 + 32909*y^20 - 65079*y^19 + 350623*y^18 - 1046531*y^17 + 1678487*y^16 - 1771438*y^15 + 2670493*y^14 - 7195176*y^13 + 14367013*y^12 - 15585945*y^11 + 7205726*y^10 - 3208396*y^9 + 37523327*y^8 - 120026000*y^7 + 201372903*y^6 - 215912334*y^5 + 156455683*y^4 - 76691304*y^3 + 25200801*y^2 - 5337738*y + 544563, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563)

$ x^{28} - 14 x^{27} + 80 x^{26} - 221 x^{25} + 397 x^{24} - 1904 x^{23} + 10213 x^{22} - 26576 x^{21} + \cdots + 544563 $ x^28 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$28$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[2, 13]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-295815184798509371659078776791937755272938949172963$ -295815184798509371659078776791937755272938949172963 $\medspace = -\,7^{13}\cdot 29^{27}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$63.47$	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:	$7^{1/2}29^{27/28}\approx 68.03282703181608$
Ramified primes:	$7$, $29$ 7, 29	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-203}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{21}a^{10}+\frac{2}{21}a^{9}+\frac{2}{21}a^{8}+\frac{1}{21}a^{7}+\frac{5}{21}a^{6}-\frac{8}{21}a^{5}+\frac{8}{21}a^{4}-\frac{2}{21}a^{3}+\frac{4}{21}a^{2}+\frac{8}{21}a-\frac{1}{7}$, $\frac{1}{21}a^{11}-\frac{2}{21}a^{9}-\frac{1}{7}a^{8}+\frac{1}{7}a^{7}+\frac{1}{7}a^{6}+\frac{1}{7}a^{5}+\frac{1}{7}a^{4}+\frac{8}{21}a^{3}+\frac{2}{21}a+\frac{2}{7}$, $\frac{1}{21}a^{12}+\frac{1}{21}a^{9}-\frac{3}{7}a^{7}+\frac{2}{7}a^{6}-\frac{2}{7}a^{5}-\frac{4}{21}a^{4}+\frac{1}{7}a^{3}+\frac{1}{7}a^{2}+\frac{8}{21}a-\frac{2}{7}$, $\frac{1}{21}a^{13}-\frac{2}{21}a^{9}+\frac{1}{7}a^{8}-\frac{3}{7}a^{7}+\frac{1}{7}a^{6}-\frac{10}{21}a^{5}+\frac{3}{7}a^{4}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}-\frac{1}{3}a+\frac{1}{7}$, $\frac{1}{63}a^{14}-\frac{1}{63}a^{13}-\frac{1}{63}a^{12}+\frac{1}{63}a^{11}+\frac{2}{21}a^{9}+\frac{10}{63}a^{8}+\frac{26}{63}a^{7}-\frac{3}{7}a^{6}-\frac{1}{7}a^{5}-\frac{16}{63}a^{4}+\frac{1}{9}a^{3}+\frac{1}{63}a^{2}+\frac{20}{63}a-\frac{2}{7}$, $\frac{1}{63}a^{15}+\frac{1}{63}a^{13}+\frac{1}{63}a^{11}-\frac{2}{63}a^{9}-\frac{1}{7}a^{8}+\frac{8}{63}a^{7}+\frac{3}{7}a^{6}-\frac{4}{9}a^{5}-\frac{1}{7}a^{4}-\frac{4}{9}a^{3}+\frac{1}{7}a^{2}-\frac{25}{63}a+\frac{1}{7}$, $\frac{1}{2205}a^{16}-\frac{8}{2205}a^{15}-\frac{4}{2205}a^{14}-\frac{2}{105}a^{13}+\frac{8}{735}a^{12}+\frac{38}{2205}a^{11}-\frac{38}{2205}a^{10}-\frac{206}{2205}a^{9}-\frac{97}{735}a^{8}-\frac{827}{2205}a^{7}+\frac{76}{441}a^{6}-\frac{928}{2205}a^{5}+\frac{94}{315}a^{4}+\frac{62}{245}a^{3}-\frac{6}{245}a^{2}-\frac{206}{2205}a-\frac{103}{245}$, $\frac{1}{2205}a^{17}+\frac{2}{2205}a^{15}-\frac{4}{2205}a^{14}+\frac{1}{735}a^{13}-\frac{10}{441}a^{12}-\frac{2}{315}a^{11}+\frac{1}{147}a^{10}+\frac{2}{35}a^{9}+\frac{34}{441}a^{8}+\frac{1079}{2205}a^{7}+\frac{319}{735}a^{6}+\frac{934}{2205}a^{5}-\frac{338}{2205}a^{4}-\frac{5}{21}a^{3}+\frac{587}{2205}a^{2}+\frac{5}{49}a+\frac{121}{245}$, $\frac{1}{6615}a^{18}+\frac{1}{6615}a^{16}+\frac{4}{6615}a^{15}+\frac{2}{315}a^{14}-\frac{148}{6615}a^{13}+\frac{137}{6615}a^{12}+\frac{4}{2205}a^{11}-\frac{151}{6615}a^{10}-\frac{359}{6615}a^{9}-\frac{139}{1323}a^{8}+\frac{233}{2205}a^{7}-\frac{2491}{6615}a^{6}-\frac{71}{1323}a^{5}-\frac{148}{315}a^{4}-\frac{2666}{6615}a^{3}+\frac{2939}{6615}a^{2}-\frac{1}{21}a-\frac{94}{245}$, $\frac{1}{6615}a^{19}+\frac{1}{6615}a^{17}+\frac{1}{6615}a^{16}-\frac{13}{2205}a^{15}-\frac{31}{6615}a^{14}+\frac{53}{6615}a^{13}+\frac{10}{441}a^{12}+\frac{10}{1323}a^{11}+\frac{2}{189}a^{10}+\frac{22}{135}a^{9}+\frac{349}{2205}a^{8}+\frac{502}{1323}a^{7}+\frac{331}{1323}a^{6}+\frac{137}{2205}a^{5}+\frac{248}{1323}a^{4}-\frac{650}{1323}a^{3}-\frac{331}{2205}a^{2}-\frac{170}{441}a-\frac{37}{245}$, $\frac{1}{19845}a^{20}-\frac{1}{19845}a^{19}-\frac{1}{6615}a^{17}+\frac{4}{19845}a^{16}-\frac{152}{19845}a^{15}-\frac{2}{315}a^{14}-\frac{184}{19845}a^{13}+\frac{16}{6615}a^{12}+\frac{79}{3969}a^{11}+\frac{34}{19845}a^{10}-\frac{436}{3969}a^{9}+\frac{838}{19845}a^{8}+\frac{418}{6615}a^{7}+\frac{2246}{19845}a^{6}+\frac{2402}{19845}a^{5}+\frac{7507}{19845}a^{4}-\frac{2239}{6615}a^{3}+\frac{964}{2835}a^{2}+\frac{94}{2205}a+\frac{12}{49}$, $\frac{1}{19845}a^{21}-\frac{1}{19845}a^{19}+\frac{1}{19845}a^{17}-\frac{1}{19845}a^{16}+\frac{157}{19845}a^{15}-\frac{26}{3969}a^{14}-\frac{13}{19845}a^{13}-\frac{20}{3969}a^{12}-\frac{121}{6615}a^{11}+\frac{62}{2835}a^{10}+\frac{677}{19845}a^{9}+\frac{3148}{19845}a^{8}+\frac{146}{405}a^{7}+\frac{551}{3969}a^{6}+\frac{719}{6615}a^{5}+\frac{5263}{19845}a^{4}-\frac{1777}{3969}a^{3}+\frac{530}{3969}a^{2}+\frac{27}{245}a+\frac{103}{245}$, $\frac{1}{99225}a^{22}-\frac{1}{99225}a^{21}+\frac{1}{99225}a^{20}+\frac{1}{19845}a^{19}-\frac{2}{99225}a^{18}+\frac{16}{99225}a^{17}-\frac{4}{19845}a^{16}-\frac{26}{11025}a^{15}+\frac{184}{33075}a^{14}-\frac{1466}{99225}a^{13}+\frac{1186}{99225}a^{12}+\frac{449}{33075}a^{11}+\frac{2021}{99225}a^{10}-\frac{8}{2025}a^{9}-\frac{458}{11025}a^{8}+\frac{8822}{99225}a^{7}+\frac{3076}{33075}a^{6}+\frac{1493}{3969}a^{5}+\frac{8774}{99225}a^{4}-\frac{257}{11025}a^{3}+\frac{925}{3969}a^{2}-\frac{386}{1575}a+\frac{62}{1225}$, $\frac{1}{99225}a^{23}+\frac{1}{99225}a^{20}-\frac{1}{14175}a^{19}-\frac{1}{99225}a^{18}-\frac{4}{99225}a^{17}+\frac{11}{99225}a^{16}+\frac{94}{14175}a^{15}-\frac{134}{99225}a^{14}-\frac{7}{405}a^{13}-\frac{752}{99225}a^{12}+\frac{1408}{99225}a^{11}+\frac{22}{14175}a^{10}-\frac{6289}{99225}a^{9}-\frac{113}{2205}a^{8}-\frac{4247}{19845}a^{7}-\frac{26162}{99225}a^{6}-\frac{1832}{33075}a^{5}+\frac{1397}{33075}a^{4}-\frac{44918}{99225}a^{3}+\frac{29117}{99225}a^{2}-\frac{4874}{11025}a-\frac{388}{1225}$, $\frac{1}{106269975}a^{24}-\frac{4}{35423325}a^{23}-\frac{166}{106269975}a^{22}+\frac{2332}{106269975}a^{21}-\frac{76}{7084665}a^{20}+\frac{6563}{106269975}a^{19}-\frac{157}{21253995}a^{18}-\frac{1693}{11807775}a^{17}+\frac{9}{145775}a^{16}-\frac{408196}{106269975}a^{15}+\frac{811061}{106269975}a^{14}-\frac{437011}{106269975}a^{13}+\frac{768511}{106269975}a^{12}-\frac{11227}{15181425}a^{11}-\frac{162541}{35423325}a^{10}+\frac{336512}{4250799}a^{9}+\frac{4243084}{35423325}a^{8}-\frac{82466}{562275}a^{7}+\frac{372584}{787185}a^{6}-\frac{2186651}{6251175}a^{5}-\frac{2314211}{11807775}a^{4}+\frac{30960841}{106269975}a^{3}-\frac{44986}{112455}a^{2}+\frac{209887}{1311975}a-\frac{18974}{145775}$, $\frac{1}{743889825}a^{25}-\frac{2}{743889825}a^{24}+\frac{1856}{743889825}a^{23}-\frac{2}{1012095}a^{22}+\frac{13612}{743889825}a^{21}-\frac{208}{15181425}a^{20}+\frac{779}{82654425}a^{19}+\frac{1090}{29755593}a^{18}-\frac{253}{4862025}a^{17}-\frac{94114}{743889825}a^{16}-\frac{262408}{148777965}a^{15}+\frac{1813087}{743889825}a^{14}-\frac{553649}{247963275}a^{13}-\frac{292829}{16530885}a^{12}+\frac{10172264}{743889825}a^{11}-\frac{5196364}{743889825}a^{10}-\frac{122541382}{743889825}a^{9}-\frac{4638496}{49592655}a^{8}+\frac{26971604}{82654425}a^{7}+\frac{274834613}{743889825}a^{6}-\frac{146107546}{743889825}a^{5}-\frac{5682547}{148777965}a^{4}-\frac{47759113}{148777965}a^{3}+\frac{28240189}{82654425}a^{2}-\frac{3264937}{9183825}a+\frac{51302}{204085}$, $\frac{1}{44\!\cdots\!25}a^{26}-\frac{13}{44\!\cdots\!25}a^{25}+\frac{14\!\cdots\!48}{32\!\cdots\!35}a^{24}-\frac{48\!\cdots\!22}{88\!\cdots\!45}a^{23}-\frac{14\!\cdots\!56}{24\!\cdots\!25}a^{22}+\frac{42\!\cdots\!54}{49\!\cdots\!25}a^{21}-\frac{45\!\cdots\!58}{44\!\cdots\!25}a^{20}-\frac{11\!\cdots\!92}{17\!\cdots\!29}a^{19}-\frac{12\!\cdots\!27}{44\!\cdots\!25}a^{18}-\frac{11\!\cdots\!52}{88\!\cdots\!45}a^{17}-\frac{96\!\cdots\!51}{49\!\cdots\!25}a^{16}+\frac{19\!\cdots\!69}{44\!\cdots\!25}a^{15}+\frac{89\!\cdots\!86}{44\!\cdots\!25}a^{14}+\frac{36\!\cdots\!73}{16\!\cdots\!75}a^{13}-\frac{10\!\cdots\!34}{88\!\cdots\!45}a^{12}+\frac{10\!\cdots\!66}{63\!\cdots\!75}a^{11}-\frac{82\!\cdots\!24}{88\!\cdots\!45}a^{10}-\frac{38\!\cdots\!48}{26\!\cdots\!25}a^{9}+\frac{43\!\cdots\!64}{14\!\cdots\!75}a^{8}-\frac{30\!\cdots\!63}{44\!\cdots\!25}a^{7}-\frac{14\!\cdots\!52}{44\!\cdots\!25}a^{6}-\frac{26\!\cdots\!16}{14\!\cdots\!75}a^{5}+\frac{53\!\cdots\!93}{44\!\cdots\!25}a^{4}+\frac{27\!\cdots\!58}{63\!\cdots\!75}a^{3}+\frac{31\!\cdots\!89}{70\!\cdots\!75}a^{2}-\frac{12\!\cdots\!21}{54\!\cdots\!25}a-\frac{19\!\cdots\!26}{60\!\cdots\!25}$, $\frac{1}{12\!\cdots\!25}a^{27}+\frac{2749}{24\!\cdots\!65}a^{26}-\frac{16\!\cdots\!61}{40\!\cdots\!75}a^{25}+\frac{35\!\cdots\!71}{13\!\cdots\!25}a^{24}+\frac{13\!\cdots\!98}{12\!\cdots\!25}a^{23}-\frac{19\!\cdots\!77}{12\!\cdots\!25}a^{22}+\frac{57\!\cdots\!46}{40\!\cdots\!75}a^{21}-\frac{27\!\cdots\!51}{13\!\cdots\!65}a^{20}-\frac{18\!\cdots\!04}{45\!\cdots\!75}a^{19}-\frac{78\!\cdots\!42}{13\!\cdots\!25}a^{18}+\frac{70\!\cdots\!57}{40\!\cdots\!75}a^{17}+\frac{82\!\cdots\!19}{13\!\cdots\!75}a^{16}-\frac{31\!\cdots\!04}{40\!\cdots\!75}a^{15}+\frac{73\!\cdots\!39}{12\!\cdots\!25}a^{14}+\frac{10\!\cdots\!39}{12\!\cdots\!25}a^{13}+\frac{16\!\cdots\!32}{26\!\cdots\!75}a^{12}-\frac{25\!\cdots\!09}{40\!\cdots\!75}a^{11}-\frac{49\!\cdots\!51}{17\!\cdots\!75}a^{10}+\frac{96\!\cdots\!69}{78\!\cdots\!15}a^{9}+\frac{17\!\cdots\!63}{12\!\cdots\!25}a^{8}+\frac{65\!\cdots\!69}{17\!\cdots\!75}a^{7}-\frac{55\!\cdots\!11}{40\!\cdots\!75}a^{6}+\frac{29\!\cdots\!73}{71\!\cdots\!25}a^{5}+\frac{43\!\cdots\!43}{24\!\cdots\!65}a^{4}-\frac{21\!\cdots\!96}{17\!\cdots\!75}a^{3}+\frac{19\!\cdots\!63}{45\!\cdots\!75}a^{2}+\frac{84\!\cdots\!65}{20\!\cdots\!51}a-\frac{33\!\cdots\!14}{16\!\cdots\!25}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/3*a^8 - 1/3*a^7 + 1/3*a^6 - 1/3*a^5 + 1/3*a^4 - 1/3*a^3 + 1/3*a^2 - 1/3*a, 1/3*a^9 - 1/3*a, 1/21*a^10 + 2/21*a^9 + 2/21*a^8 + 1/21*a^7 + 5/21*a^6 - 8/21*a^5 + 8/21*a^4 - 2/21*a^3 + 4/21*a^2 + 8/21*a - 1/7, 1/21*a^11 - 2/21*a^9 - 1/7*a^8 + 1/7*a^7 + 1/7*a^6 + 1/7*a^5 + 1/7*a^4 + 8/21*a^3 + 2/21*a + 2/7, 1/21*a^12 + 1/21*a^9 - 3/7*a^7 + 2/7*a^6 - 2/7*a^5 - 4/21*a^4 + 1/7*a^3 + 1/7*a^2 + 8/21*a - 2/7, 1/21*a^13 - 2/21*a^9 + 1/7*a^8 - 3/7*a^7 + 1/7*a^6 - 10/21*a^5 + 3/7*a^4 - 3/7*a^3 - 1/7*a^2 - 1/3*a + 1/7, 1/63*a^14 - 1/63*a^13 - 1/63*a^12 + 1/63*a^11 + 2/21*a^9 + 10/63*a^8 + 26/63*a^7 - 3/7*a^6 - 1/7*a^5 - 16/63*a^4 + 1/9*a^3 + 1/63*a^2 + 20/63*a - 2/7, 1/63*a^15 + 1/63*a^13 + 1/63*a^11 - 2/63*a^9 - 1/7*a^8 + 8/63*a^7 + 3/7*a^6 - 4/9*a^5 - 1/7*a^4 - 4/9*a^3 + 1/7*a^2 - 25/63*a + 1/7, 1/2205*a^16 - 8/2205*a^15 - 4/2205*a^14 - 2/105*a^13 + 8/735*a^12 + 38/2205*a^11 - 38/2205*a^10 - 206/2205*a^9 - 97/735*a^8 - 827/2205*a^7 + 76/441*a^6 - 928/2205*a^5 + 94/315*a^4 + 62/245*a^3 - 6/245*a^2 - 206/2205*a - 103/245, 1/2205*a^17 + 2/2205*a^15 - 4/2205*a^14 + 1/735*a^13 - 10/441*a^12 - 2/315*a^11 + 1/147*a^10 + 2/35*a^9 + 34/441*a^8 + 1079/2205*a^7 + 319/735*a^6 + 934/2205*a^5 - 338/2205*a^4 - 5/21*a^3 + 587/2205*a^2 + 5/49*a + 121/245, 1/6615*a^18 + 1/6615*a^16 + 4/6615*a^15 + 2/315*a^14 - 148/6615*a^13 + 137/6615*a^12 + 4/2205*a^11 - 151/6615*a^10 - 359/6615*a^9 - 139/1323*a^8 + 233/2205*a^7 - 2491/6615*a^6 - 71/1323*a^5 - 148/315*a^4 - 2666/6615*a^3 + 2939/6615*a^2 - 1/21*a - 94/245, 1/6615*a^19 + 1/6615*a^17 + 1/6615*a^16 - 13/2205*a^15 - 31/6615*a^14 + 53/6615*a^13 + 10/441*a^12 + 10/1323*a^11 + 2/189*a^10 + 22/135*a^9 + 349/2205*a^8 + 502/1323*a^7 + 331/1323*a^6 + 137/2205*a^5 + 248/1323*a^4 - 650/1323*a^3 - 331/2205*a^2 - 170/441*a - 37/245, 1/19845*a^20 - 1/19845*a^19 - 1/6615*a^17 + 4/19845*a^16 - 152/19845*a^15 - 2/315*a^14 - 184/19845*a^13 + 16/6615*a^12 + 79/3969*a^11 + 34/19845*a^10 - 436/3969*a^9 + 838/19845*a^8 + 418/6615*a^7 + 2246/19845*a^6 + 2402/19845*a^5 + 7507/19845*a^4 - 2239/6615*a^3 + 964/2835*a^2 + 94/2205*a + 12/49, 1/19845*a^21 - 1/19845*a^19 + 1/19845*a^17 - 1/19845*a^16 + 157/19845*a^15 - 26/3969*a^14 - 13/19845*a^13 - 20/3969*a^12 - 121/6615*a^11 + 62/2835*a^10 + 677/19845*a^9 + 3148/19845*a^8 + 146/405*a^7 + 551/3969*a^6 + 719/6615*a^5 + 5263/19845*a^4 - 1777/3969*a^3 + 530/3969*a^2 + 27/245*a + 103/245, 1/99225*a^22 - 1/99225*a^21 + 1/99225*a^20 + 1/19845*a^19 - 2/99225*a^18 + 16/99225*a^17 - 4/19845*a^16 - 26/11025*a^15 + 184/33075*a^14 - 1466/99225*a^13 + 1186/99225*a^12 + 449/33075*a^11 + 2021/99225*a^10 - 8/2025*a^9 - 458/11025*a^8 + 8822/99225*a^7 + 3076/33075*a^6 + 1493/3969*a^5 + 8774/99225*a^4 - 257/11025*a^3 + 925/3969*a^2 - 386/1575*a + 62/1225, 1/99225*a^23 + 1/99225*a^20 - 1/14175*a^19 - 1/99225*a^18 - 4/99225*a^17 + 11/99225*a^16 + 94/14175*a^15 - 134/99225*a^14 - 7/405*a^13 - 752/99225*a^12 + 1408/99225*a^11 + 22/14175*a^10 - 6289/99225*a^9 - 113/2205*a^8 - 4247/19845*a^7 - 26162/99225*a^6 - 1832/33075*a^5 + 1397/33075*a^4 - 44918/99225*a^3 + 29117/99225*a^2 - 4874/11025*a - 388/1225, 1/106269975*a^24 - 4/35423325*a^23 - 166/106269975*a^22 + 2332/106269975*a^21 - 76/7084665*a^20 + 6563/106269975*a^19 - 157/21253995*a^18 - 1693/11807775*a^17 + 9/145775*a^16 - 408196/106269975*a^15 + 811061/106269975*a^14 - 437011/106269975*a^13 + 768511/106269975*a^12 - 11227/15181425*a^11 - 162541/35423325*a^10 + 336512/4250799*a^9 + 4243084/35423325*a^8 - 82466/562275*a^7 + 372584/787185*a^6 - 2186651/6251175*a^5 - 2314211/11807775*a^4 + 30960841/106269975*a^3 - 44986/112455*a^2 + 209887/1311975*a - 18974/145775, 1/743889825*a^25 - 2/743889825*a^24 + 1856/743889825*a^23 - 2/1012095*a^22 + 13612/743889825*a^21 - 208/15181425*a^20 + 779/82654425*a^19 + 1090/29755593*a^18 - 253/4862025*a^17 - 94114/743889825*a^16 - 262408/148777965*a^15 + 1813087/743889825*a^14 - 553649/247963275*a^13 - 292829/16530885*a^12 + 10172264/743889825*a^11 - 5196364/743889825*a^10 - 122541382/743889825*a^9 - 4638496/49592655*a^8 + 26971604/82654425*a^7 + 274834613/743889825*a^6 - 146107546/743889825*a^5 - 5682547/148777965*a^4 - 47759113/148777965*a^3 + 28240189/82654425*a^2 - 3264937/9183825*a + 51302/204085, 1/444288725390527299426348225*a^26 - 13/444288725390527299426348225*a^25 + 14929591166253548/3291027595485387403158135*a^24 - 4837187537866149422/88857745078105459885269645*a^23 - 1424962676308954556/2454633841936614913957725*a^22 + 428549006788656483554/49365413932280811047372025*a^21 - 4519612930980753080758/444288725390527299426348225*a^20 - 1190874478291723993592/17771549015621091977053929*a^19 - 12637344902030300735327/444288725390527299426348225*a^18 - 11436094070821868902652/88857745078105459885269645*a^17 - 9613232122789449724951/49365413932280811047372025*a^16 + 1928884623113938311527969/444288725390527299426348225*a^15 + 894390855275860368662986/444288725390527299426348225*a^14 + 362957771903177040569473/16455137977426937015790675*a^13 - 1082615351607159116520134/88857745078105459885269645*a^12 + 1003682099084240376214066/63469817912932471346621175*a^11 - 823165327470834689084524/88857745078105459885269645*a^10 - 3844990545137738083269848/26134630905325135260373425*a^9 + 4327098881031525015437164/148096241796842433142116075*a^8 - 3033684371714490356905663/444288725390527299426348225*a^7 - 147753220879114397096222252/444288725390527299426348225*a^6 - 26202055523342651437050316/148096241796842433142116075*a^5 + 53868902129361448885785893/444288725390527299426348225*a^4 + 2757307838320837965014458/63469817912932471346621175*a^3 + 3177156674015960535679489/7052201990325830149624575*a^2 - 1272086132929636374643721/5485045992475645671930225*a - 193332964091403385094826/609449554719516185770025, 1/12225492856571139698314824107325*a^27 + 2749/2445098571314227939662964821465*a^26 - 1642364868824682848261/4075164285523713232771608035775*a^25 + 3533105636869562842271/1358388095174571077590536011925*a^24 + 13020465448257778136217598/12225492856571139698314824107325*a^23 - 19247602425155918307387877/12225492856571139698314824107325*a^22 + 57621620900428029924574946/4075164285523713232771608035775*a^21 - 278675052925011521893351/13508831885713966517474943765*a^20 - 18103439270698948299637304/452796031724857025863512003975*a^19 - 78813956366367572557945942/1358388095174571077590536011925*a^18 + 700105620071893632025134757/4075164285523713232771608035775*a^17 + 826794563041247787095819/13131571274512502361240412575*a^16 - 31677702668318389338773876204/4075164285523713232771608035775*a^15 + 73339012450063484717084552339/12225492856571139698314824107325*a^14 + 10799170935428610825526082839/12225492856571139698314824107325*a^13 + 163456612000985413318118332/260116869288747653155634555475*a^12 - 25643936923199007905512015109/4075164285523713232771608035775*a^11 - 496256221091575093596383851/1746498979510162814044974872475*a^10 + 968247976242895252470974369/7862053283968578584125288815*a^9 + 1704625582763321274733874959163/12225492856571139698314824107325*a^8 + 659254194517136852085636820369/1746498979510162814044974872475*a^7 - 551606572118155316990969295611/4075164285523713232771608035775*a^6 + 295326609152010477372270587573/719146638621831746959695535725*a^5 + 436447747898837174994034202843/2445098571314227939662964821465*a^4 - 214124629504963747745523656096/1746498979510162814044974872475*a^3 + 199129207888007937190047152963/452796031724857025863512003975*a^2 + 841953186457373010238115365/2012426807666031226060053351*a - 3340135953342914918791303414/16770223397216926883833777925

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$3$, $5$

Class group and class number

$C_{7}$, which has order $7$ (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:	$14$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -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:	$\frac{488611}{510111886095}a^{26}-\frac{6351943}{510111886095}a^{25}+\frac{13080549607}{206595313868475}a^{24}-\frac{9446583178}{68865104622825}a^{23}+\frac{38448811007}{206595313868475}a^{22}-\frac{62539442977}{41319062773695}a^{21}+\frac{557701827568}{68865104622825}a^{20}-\frac{194357120702}{12152665521675}a^{19}+\frac{1704112735522}{206595313868475}a^{18}-\frac{181908514049}{4591006974855}a^{17}+\frac{6619643874548}{22955034874275}a^{16}-\frac{140082159965218}{206595313868475}a^{15}+\frac{1631862790736}{2430533104335}a^{14}-\frac{85419870598606}{206595313868475}a^{13}+\frac{314590230468916}{206595313868475}a^{12}-\frac{29376148141217}{5902723253385}a^{11}+\frac{512019105626036}{68865104622825}a^{10}-\frac{627357685491028}{206595313868475}a^{9}-\frac{202792705471673}{68865104622825}a^{8}-\frac{4623655332811}{1530335658285}a^{7}+\frac{814389001367198}{22955034874275}a^{6}-\frac{15\!\cdots\!19}{206595313868475}a^{5}+\frac{19\!\cdots\!52}{22955034874275}a^{4}-\frac{309066381422812}{5902723253385}a^{3}+\frac{1468490012419}{72873126585}a^{2}-\frac{12501469099423}{2550559430475}a+\frac{1630475588459}{283395492275}$, $\frac{28\!\cdots\!39}{90\!\cdots\!95}a^{27}-\frac{61\!\cdots\!09}{13\!\cdots\!25}a^{26}+\frac{18\!\cdots\!84}{71\!\cdots\!25}a^{25}-\frac{17\!\cdots\!69}{23\!\cdots\!75}a^{24}+\frac{15\!\cdots\!17}{12\!\cdots\!25}a^{23}-\frac{14\!\cdots\!44}{24\!\cdots\!65}a^{22}+\frac{23\!\cdots\!81}{71\!\cdots\!75}a^{21}-\frac{10\!\cdots\!73}{12\!\cdots\!25}a^{20}+\frac{25\!\cdots\!72}{24\!\cdots\!65}a^{19}-\frac{79\!\cdots\!34}{40\!\cdots\!75}a^{18}+\frac{32\!\cdots\!04}{28\!\cdots\!75}a^{17}-\frac{85\!\cdots\!04}{24\!\cdots\!25}a^{16}+\frac{66\!\cdots\!17}{12\!\cdots\!25}a^{15}-\frac{65\!\cdots\!06}{12\!\cdots\!25}a^{14}+\frac{98\!\cdots\!78}{12\!\cdots\!25}a^{13}-\frac{28\!\cdots\!67}{12\!\cdots\!25}a^{12}+\frac{63\!\cdots\!39}{13\!\cdots\!25}a^{11}-\frac{85\!\cdots\!76}{17\!\cdots\!75}a^{10}+\frac{14\!\cdots\!26}{81\!\cdots\!55}a^{9}-\frac{51\!\cdots\!19}{90\!\cdots\!95}a^{8}+\frac{23\!\cdots\!13}{19\!\cdots\!75}a^{7}-\frac{28\!\cdots\!28}{71\!\cdots\!25}a^{6}+\frac{81\!\cdots\!23}{12\!\cdots\!25}a^{5}-\frac{80\!\cdots\!33}{12\!\cdots\!25}a^{4}+\frac{25\!\cdots\!83}{58\!\cdots\!25}a^{3}-\frac{25\!\cdots\!26}{13\!\cdots\!25}a^{2}+\frac{74\!\cdots\!64}{15\!\cdots\!25}a-\frac{11\!\cdots\!66}{16\!\cdots\!25}$, $\frac{28\!\cdots\!39}{90\!\cdots\!95}a^{27}-\frac{29\!\cdots\!94}{71\!\cdots\!75}a^{26}+\frac{24\!\cdots\!37}{12\!\cdots\!25}a^{25}-\frac{52\!\cdots\!81}{12\!\cdots\!25}a^{24}+\frac{41\!\cdots\!31}{67\!\cdots\!25}a^{23}-\frac{12\!\cdots\!41}{26\!\cdots\!75}a^{22}+\frac{31\!\cdots\!26}{12\!\cdots\!25}a^{21}-\frac{36\!\cdots\!03}{71\!\cdots\!25}a^{20}+\frac{33\!\cdots\!44}{12\!\cdots\!25}a^{19}-\frac{16\!\cdots\!57}{12\!\cdots\!25}a^{18}+\frac{13\!\cdots\!29}{15\!\cdots\!25}a^{17}-\frac{37\!\cdots\!88}{17\!\cdots\!75}a^{16}+\frac{17\!\cdots\!73}{81\!\cdots\!55}a^{15}-\frac{38\!\cdots\!87}{27\!\cdots\!85}a^{14}+\frac{57\!\cdots\!79}{12\!\cdots\!25}a^{13}-\frac{18\!\cdots\!26}{12\!\cdots\!25}a^{12}+\frac{28\!\cdots\!89}{12\!\cdots\!25}a^{11}-\frac{18\!\cdots\!44}{17\!\cdots\!75}a^{10}-\frac{11\!\cdots\!23}{12\!\cdots\!25}a^{9}-\frac{22\!\cdots\!97}{40\!\cdots\!75}a^{8}+\frac{72\!\cdots\!63}{64\!\cdots\!25}a^{7}-\frac{29\!\cdots\!81}{12\!\cdots\!25}a^{6}+\frac{67\!\cdots\!19}{24\!\cdots\!65}a^{5}-\frac{20\!\cdots\!02}{12\!\cdots\!25}a^{4}+\frac{63\!\cdots\!62}{17\!\cdots\!75}a^{3}+\frac{10\!\cdots\!67}{54\!\cdots\!77}a^{2}-\frac{17\!\cdots\!96}{15\!\cdots\!25}a+\frac{18\!\cdots\!64}{92\!\cdots\!25}$, $\frac{18\!\cdots\!51}{12\!\cdots\!25}a^{27}-\frac{28\!\cdots\!13}{12\!\cdots\!25}a^{26}+\frac{61\!\cdots\!81}{40\!\cdots\!75}a^{25}-\frac{66\!\cdots\!29}{13\!\cdots\!25}a^{24}+\frac{39\!\cdots\!26}{40\!\cdots\!75}a^{23}-\frac{14\!\cdots\!61}{40\!\cdots\!75}a^{22}+\frac{86\!\cdots\!21}{45\!\cdots\!75}a^{21}-\frac{24\!\cdots\!83}{40\!\cdots\!75}a^{20}+\frac{11\!\cdots\!04}{12\!\cdots\!25}a^{19}-\frac{16\!\cdots\!08}{12\!\cdots\!25}a^{18}+\frac{78\!\cdots\!06}{12\!\cdots\!25}a^{17}-\frac{13\!\cdots\!32}{58\!\cdots\!25}a^{16}+\frac{10\!\cdots\!63}{24\!\cdots\!65}a^{15}-\frac{12\!\cdots\!94}{24\!\cdots\!65}a^{14}+\frac{24\!\cdots\!56}{40\!\cdots\!75}a^{13}-\frac{18\!\cdots\!34}{12\!\cdots\!25}a^{12}+\frac{41\!\cdots\!47}{12\!\cdots\!25}a^{11}-\frac{80\!\cdots\!89}{17\!\cdots\!75}a^{10}+\frac{22\!\cdots\!07}{81\!\cdots\!55}a^{9}-\frac{10\!\cdots\!82}{24\!\cdots\!65}a^{8}+\frac{74\!\cdots\!71}{12\!\cdots\!25}a^{7}-\frac{10\!\cdots\!76}{40\!\cdots\!75}a^{6}+\frac{63\!\cdots\!29}{12\!\cdots\!25}a^{5}-\frac{75\!\cdots\!64}{12\!\cdots\!25}a^{4}+\frac{27\!\cdots\!06}{58\!\cdots\!25}a^{3}-\frac{20\!\cdots\!93}{88\!\cdots\!25}a^{2}+\frac{95\!\cdots\!96}{15\!\cdots\!25}a-\frac{67\!\cdots\!51}{88\!\cdots\!75}$, $\frac{18\!\cdots\!53}{24\!\cdots\!65}a^{27}-\frac{13\!\cdots\!69}{13\!\cdots\!25}a^{26}+\frac{62\!\cdots\!64}{12\!\cdots\!25}a^{25}-\frac{14\!\cdots\!42}{12\!\cdots\!25}a^{24}+\frac{45\!\cdots\!59}{23\!\cdots\!75}a^{23}-\frac{32\!\cdots\!86}{26\!\cdots\!75}a^{22}+\frac{79\!\cdots\!14}{12\!\cdots\!25}a^{21}-\frac{57\!\cdots\!73}{40\!\cdots\!75}a^{20}+\frac{17\!\cdots\!29}{14\!\cdots\!45}a^{19}-\frac{11\!\cdots\!39}{30\!\cdots\!65}a^{18}+\frac{27\!\cdots\!26}{12\!\cdots\!25}a^{17}-\frac{10\!\cdots\!61}{17\!\cdots\!75}a^{16}+\frac{17\!\cdots\!32}{23\!\cdots\!75}a^{15}-\frac{78\!\cdots\!27}{12\!\cdots\!25}a^{14}+\frac{33\!\cdots\!76}{24\!\cdots\!65}a^{13}-\frac{56\!\cdots\!81}{13\!\cdots\!25}a^{12}+\frac{17\!\cdots\!61}{24\!\cdots\!65}a^{11}-\frac{61\!\cdots\!78}{11\!\cdots\!65}a^{10}+\frac{31\!\cdots\!58}{12\!\cdots\!25}a^{9}-\frac{19\!\cdots\!16}{12\!\cdots\!25}a^{8}+\frac{15\!\cdots\!23}{58\!\cdots\!25}a^{7}-\frac{16\!\cdots\!03}{24\!\cdots\!65}a^{6}+\frac{80\!\cdots\!06}{87\!\cdots\!75}a^{5}-\frac{10\!\cdots\!78}{13\!\cdots\!25}a^{4}+\frac{74\!\cdots\!52}{17\!\cdots\!75}a^{3}-\frac{42\!\cdots\!21}{30\!\cdots\!65}a^{2}+\frac{93\!\cdots\!21}{30\!\cdots\!65}a-\frac{38\!\cdots\!29}{16\!\cdots\!25}$, $\frac{48\!\cdots\!42}{12\!\cdots\!25}a^{27}-\frac{24\!\cdots\!98}{40\!\cdots\!75}a^{26}+\frac{17\!\cdots\!07}{45\!\cdots\!75}a^{25}-\frac{79\!\cdots\!09}{64\!\cdots\!75}a^{24}+\frac{58\!\cdots\!61}{24\!\cdots\!65}a^{23}-\frac{61\!\cdots\!23}{71\!\cdots\!75}a^{22}+\frac{59\!\cdots\!78}{12\!\cdots\!25}a^{21}-\frac{73\!\cdots\!25}{48\!\cdots\!93}a^{20}+\frac{27\!\cdots\!43}{12\!\cdots\!25}a^{19}-\frac{38\!\cdots\!36}{12\!\cdots\!25}a^{18}+\frac{22\!\cdots\!77}{14\!\cdots\!45}a^{17}-\frac{99\!\cdots\!13}{17\!\cdots\!75}a^{16}+\frac{43\!\cdots\!44}{40\!\cdots\!75}a^{15}-\frac{27\!\cdots\!61}{24\!\cdots\!65}a^{14}+\frac{80\!\cdots\!93}{64\!\cdots\!75}a^{13}-\frac{43\!\cdots\!81}{12\!\cdots\!25}a^{12}+\frac{69\!\cdots\!56}{81\!\cdots\!55}a^{11}-\frac{18\!\cdots\!19}{17\!\cdots\!75}a^{10}+\frac{77\!\cdots\!02}{16\!\cdots\!31}a^{9}+\frac{15\!\cdots\!39}{12\!\cdots\!25}a^{8}+\frac{88\!\cdots\!31}{58\!\cdots\!25}a^{7}-\frac{27\!\cdots\!84}{40\!\cdots\!75}a^{6}+\frac{57\!\cdots\!16}{45\!\cdots\!75}a^{5}-\frac{16\!\cdots\!04}{12\!\cdots\!25}a^{4}+\frac{51\!\cdots\!52}{58\!\cdots\!25}a^{3}-\frac{84\!\cdots\!37}{26\!\cdots\!75}a^{2}+\frac{16\!\cdots\!42}{33\!\cdots\!85}a-\frac{22\!\cdots\!41}{67\!\cdots\!17}$, $\frac{62\!\cdots\!81}{40\!\cdots\!75}a^{27}+\frac{14\!\cdots\!47}{12\!\cdots\!25}a^{26}-\frac{21\!\cdots\!24}{71\!\cdots\!25}a^{25}+\frac{86\!\cdots\!08}{48\!\cdots\!93}a^{24}-\frac{15\!\cdots\!29}{40\!\cdots\!75}a^{23}+\frac{13\!\cdots\!37}{35\!\cdots\!87}a^{22}-\frac{45\!\cdots\!46}{12\!\cdots\!25}a^{21}+\frac{30\!\cdots\!34}{13\!\cdots\!25}a^{20}-\frac{57\!\cdots\!03}{12\!\cdots\!25}a^{19}+\frac{32\!\cdots\!99}{13\!\cdots\!25}a^{18}-\frac{12\!\cdots\!48}{12\!\cdots\!25}a^{17}+\frac{13\!\cdots\!92}{17\!\cdots\!75}a^{16}-\frac{23\!\cdots\!42}{12\!\cdots\!25}a^{15}+\frac{29\!\cdots\!74}{13\!\cdots\!25}a^{14}-\frac{70\!\cdots\!48}{42\!\cdots\!75}a^{13}+\frac{53\!\cdots\!47}{12\!\cdots\!25}a^{12}-\frac{17\!\cdots\!19}{12\!\cdots\!25}a^{11}+\frac{39\!\cdots\!12}{17\!\cdots\!75}a^{10}-\frac{56\!\cdots\!38}{42\!\cdots\!45}a^{9}-\frac{10\!\cdots\!71}{40\!\cdots\!75}a^{8}-\frac{96\!\cdots\!22}{24\!\cdots\!25}a^{7}+\frac{11\!\cdots\!46}{12\!\cdots\!25}a^{6}-\frac{53\!\cdots\!69}{24\!\cdots\!65}a^{5}+\frac{33\!\cdots\!54}{12\!\cdots\!25}a^{4}-\frac{35\!\cdots\!58}{16\!\cdots\!95}a^{3}+\frac{14\!\cdots\!94}{13\!\cdots\!25}a^{2}-\frac{16\!\cdots\!42}{56\!\cdots\!75}a+\frac{95\!\cdots\!06}{16\!\cdots\!25}$, $\frac{62\!\cdots\!81}{40\!\cdots\!75}a^{27}-\frac{65\!\cdots\!08}{12\!\cdots\!25}a^{26}+\frac{24\!\cdots\!41}{45\!\cdots\!75}a^{25}-\frac{12\!\cdots\!42}{48\!\cdots\!93}a^{24}+\frac{73\!\cdots\!63}{12\!\cdots\!25}a^{23}-\frac{14\!\cdots\!14}{12\!\cdots\!25}a^{22}+\frac{84\!\cdots\!83}{12\!\cdots\!25}a^{21}-\frac{84\!\cdots\!84}{26\!\cdots\!75}a^{20}+\frac{83\!\cdots\!97}{12\!\cdots\!25}a^{19}-\frac{58\!\cdots\!04}{87\!\cdots\!75}a^{18}+\frac{26\!\cdots\!73}{12\!\cdots\!25}a^{17}-\frac{20\!\cdots\!58}{17\!\cdots\!75}a^{16}+\frac{33\!\cdots\!52}{11\!\cdots\!75}a^{15}-\frac{44\!\cdots\!52}{12\!\cdots\!25}a^{14}+\frac{13\!\cdots\!04}{40\!\cdots\!75}a^{13}-\frac{87\!\cdots\!27}{12\!\cdots\!25}a^{12}+\frac{30\!\cdots\!28}{15\!\cdots\!25}a^{11}-\frac{13\!\cdots\!79}{38\!\cdots\!55}a^{10}+\frac{31\!\cdots\!88}{12\!\cdots\!25}a^{9}-\frac{43\!\cdots\!93}{17\!\cdots\!65}a^{8}+\frac{20\!\cdots\!07}{17\!\cdots\!75}a^{7}-\frac{54\!\cdots\!52}{40\!\cdots\!75}a^{6}+\frac{80\!\cdots\!12}{24\!\cdots\!65}a^{5}-\frac{18\!\cdots\!54}{40\!\cdots\!75}a^{4}+\frac{95\!\cdots\!68}{24\!\cdots\!25}a^{3}-\frac{16\!\cdots\!83}{79\!\cdots\!25}a^{2}+\frac{10\!\cdots\!99}{15\!\cdots\!25}a-\frac{11\!\cdots\!84}{88\!\cdots\!75}$, $\frac{10\!\cdots\!48}{12\!\cdots\!25}a^{27}-\frac{30\!\cdots\!82}{24\!\cdots\!65}a^{26}+\frac{84\!\cdots\!87}{12\!\cdots\!25}a^{25}-\frac{22\!\cdots\!56}{12\!\cdots\!25}a^{24}+\frac{38\!\cdots\!89}{12\!\cdots\!25}a^{23}-\frac{13\!\cdots\!41}{81\!\cdots\!55}a^{22}+\frac{10\!\cdots\!59}{12\!\cdots\!25}a^{21}-\frac{53\!\cdots\!63}{24\!\cdots\!65}a^{20}+\frac{12\!\cdots\!12}{48\!\cdots\!93}a^{19}-\frac{34\!\cdots\!84}{64\!\cdots\!75}a^{18}+\frac{37\!\cdots\!38}{12\!\cdots\!25}a^{17}-\frac{15\!\cdots\!73}{17\!\cdots\!75}a^{16}+\frac{16\!\cdots\!76}{12\!\cdots\!25}a^{15}-\frac{32\!\cdots\!16}{24\!\cdots\!65}a^{14}+\frac{29\!\cdots\!29}{13\!\cdots\!25}a^{13}-\frac{73\!\cdots\!23}{12\!\cdots\!25}a^{12}+\frac{47\!\cdots\!31}{40\!\cdots\!75}a^{11}-\frac{67\!\cdots\!59}{58\!\cdots\!25}a^{10}+\frac{64\!\cdots\!76}{14\!\cdots\!45}a^{9}-\frac{31\!\cdots\!48}{12\!\cdots\!25}a^{8}+\frac{58\!\cdots\!97}{17\!\cdots\!75}a^{7}-\frac{24\!\cdots\!37}{24\!\cdots\!65}a^{6}+\frac{19\!\cdots\!32}{12\!\cdots\!25}a^{5}-\frac{66\!\cdots\!03}{40\!\cdots\!75}a^{4}+\frac{19\!\cdots\!61}{17\!\cdots\!75}a^{3}-\frac{71\!\cdots\!48}{13\!\cdots\!25}a^{2}+\frac{24\!\cdots\!44}{15\!\cdots\!25}a-\frac{32\!\cdots\!03}{12\!\cdots\!75}$, $\frac{30\!\cdots\!13}{12\!\cdots\!25}a^{27}-\frac{47\!\cdots\!31}{13\!\cdots\!25}a^{26}+\frac{12\!\cdots\!87}{64\!\cdots\!75}a^{25}-\frac{63\!\cdots\!93}{12\!\cdots\!25}a^{24}+\frac{36\!\cdots\!52}{40\!\cdots\!75}a^{23}-\frac{56\!\cdots\!32}{12\!\cdots\!25}a^{22}+\frac{60\!\cdots\!99}{24\!\cdots\!65}a^{21}-\frac{25\!\cdots\!62}{40\!\cdots\!75}a^{20}+\frac{29\!\cdots\!79}{40\!\cdots\!75}a^{19}-\frac{18\!\cdots\!18}{12\!\cdots\!25}a^{18}+\frac{20\!\cdots\!48}{24\!\cdots\!65}a^{17}-\frac{36\!\cdots\!97}{14\!\cdots\!17}a^{16}+\frac{46\!\cdots\!34}{12\!\cdots\!25}a^{15}-\frac{92\!\cdots\!62}{24\!\cdots\!65}a^{14}+\frac{81\!\cdots\!12}{13\!\cdots\!25}a^{13}-\frac{20\!\cdots\!07}{12\!\cdots\!25}a^{12}+\frac{13\!\cdots\!94}{40\!\cdots\!75}a^{11}-\frac{64\!\cdots\!01}{19\!\cdots\!75}a^{10}+\frac{48\!\cdots\!62}{40\!\cdots\!75}a^{9}-\frac{37\!\cdots\!87}{71\!\cdots\!25}a^{8}+\frac{54\!\cdots\!21}{58\!\cdots\!25}a^{7}-\frac{70\!\cdots\!68}{24\!\cdots\!65}a^{6}+\frac{62\!\cdots\!87}{13\!\cdots\!25}a^{5}-\frac{83\!\cdots\!29}{18\!\cdots\!59}a^{4}+\frac{17\!\cdots\!54}{58\!\cdots\!25}a^{3}-\frac{17\!\cdots\!69}{13\!\cdots\!25}a^{2}+\frac{17\!\cdots\!19}{50\!\cdots\!75}a-\frac{92\!\cdots\!42}{16\!\cdots\!25}$, $\frac{11\!\cdots\!33}{71\!\cdots\!75}a^{27}-\frac{52\!\cdots\!49}{24\!\cdots\!65}a^{26}+\frac{14\!\cdots\!02}{12\!\cdots\!25}a^{25}-\frac{39\!\cdots\!14}{12\!\cdots\!25}a^{24}+\frac{69\!\cdots\!24}{12\!\cdots\!25}a^{23}-\frac{11\!\cdots\!48}{40\!\cdots\!75}a^{22}+\frac{75\!\cdots\!30}{48\!\cdots\!93}a^{21}-\frac{47\!\cdots\!13}{12\!\cdots\!25}a^{20}+\frac{61\!\cdots\!52}{13\!\cdots\!25}a^{19}-\frac{11\!\cdots\!01}{12\!\cdots\!25}a^{18}+\frac{64\!\cdots\!11}{12\!\cdots\!25}a^{17}-\frac{60\!\cdots\!94}{38\!\cdots\!55}a^{16}+\frac{29\!\cdots\!49}{12\!\cdots\!25}a^{15}-\frac{32\!\cdots\!57}{13\!\cdots\!25}a^{14}+\frac{45\!\cdots\!26}{12\!\cdots\!25}a^{13}-\frac{43\!\cdots\!87}{40\!\cdots\!75}a^{12}+\frac{28\!\cdots\!12}{13\!\cdots\!25}a^{11}-\frac{52\!\cdots\!99}{24\!\cdots\!25}a^{10}+\frac{53\!\cdots\!69}{71\!\cdots\!25}a^{9}-\frac{25\!\cdots\!31}{81\!\cdots\!55}a^{8}+\frac{10\!\cdots\!16}{17\!\cdots\!75}a^{7}-\frac{21\!\cdots\!39}{12\!\cdots\!25}a^{6}+\frac{74\!\cdots\!87}{26\!\cdots\!75}a^{5}-\frac{35\!\cdots\!91}{12\!\cdots\!25}a^{4}+\frac{33\!\cdots\!29}{17\!\cdots\!75}a^{3}-\frac{37\!\cdots\!52}{45\!\cdots\!75}a^{2}+\frac{66\!\cdots\!51}{30\!\cdots\!65}a-\frac{51\!\cdots\!34}{16\!\cdots\!25}$, $\frac{50\!\cdots\!03}{40\!\cdots\!75}a^{27}-\frac{80\!\cdots\!43}{48\!\cdots\!93}a^{26}+\frac{42\!\cdots\!80}{48\!\cdots\!93}a^{25}-\frac{25\!\cdots\!77}{12\!\cdots\!25}a^{24}+\frac{41\!\cdots\!79}{12\!\cdots\!25}a^{23}-\frac{51\!\cdots\!29}{24\!\cdots\!65}a^{22}+\frac{13\!\cdots\!79}{12\!\cdots\!25}a^{21}-\frac{30\!\cdots\!48}{12\!\cdots\!25}a^{20}+\frac{27\!\cdots\!54}{12\!\cdots\!25}a^{19}-\frac{26\!\cdots\!24}{40\!\cdots\!75}a^{18}+\frac{94\!\cdots\!34}{24\!\cdots\!65}a^{17}-\frac{59\!\cdots\!59}{58\!\cdots\!25}a^{16}+\frac{28\!\cdots\!03}{21\!\cdots\!25}a^{15}-\frac{14\!\cdots\!89}{12\!\cdots\!25}a^{14}+\frac{29\!\cdots\!22}{12\!\cdots\!25}a^{13}-\frac{18\!\cdots\!89}{26\!\cdots\!75}a^{12}+\frac{30\!\cdots\!82}{24\!\cdots\!65}a^{11}-\frac{17\!\cdots\!61}{17\!\cdots\!75}a^{10}+\frac{12\!\cdots\!22}{81\!\cdots\!55}a^{9}-\frac{60\!\cdots\!02}{21\!\cdots\!25}a^{8}+\frac{77\!\cdots\!74}{17\!\cdots\!75}a^{7}-\frac{14\!\cdots\!23}{12\!\cdots\!25}a^{6}+\frac{20\!\cdots\!48}{12\!\cdots\!25}a^{5}-\frac{18\!\cdots\!31}{12\!\cdots\!25}a^{4}+\frac{37\!\cdots\!09}{43\!\cdots\!95}a^{3}-\frac{92\!\cdots\!55}{28\!\cdots\!83}a^{2}+\frac{69\!\cdots\!02}{88\!\cdots\!25}a-\frac{15\!\cdots\!67}{16\!\cdots\!25}$, $\frac{14\!\cdots\!38}{12\!\cdots\!25}a^{27}-\frac{20\!\cdots\!73}{12\!\cdots\!25}a^{26}+\frac{23\!\cdots\!44}{24\!\cdots\!65}a^{25}-\frac{31\!\cdots\!83}{12\!\cdots\!25}a^{24}+\frac{21\!\cdots\!14}{48\!\cdots\!93}a^{23}-\frac{26\!\cdots\!73}{12\!\cdots\!25}a^{22}+\frac{14\!\cdots\!66}{12\!\cdots\!25}a^{21}-\frac{75\!\cdots\!69}{24\!\cdots\!65}a^{20}+\frac{13\!\cdots\!48}{40\!\cdots\!75}a^{19}-\frac{78\!\cdots\!44}{12\!\cdots\!25}a^{18}+\frac{16\!\cdots\!28}{40\!\cdots\!75}a^{17}-\frac{14\!\cdots\!01}{11\!\cdots\!65}a^{16}+\frac{44\!\cdots\!56}{24\!\cdots\!65}a^{15}-\frac{11\!\cdots\!33}{70\!\cdots\!85}a^{14}+\frac{18\!\cdots\!97}{71\!\cdots\!25}a^{13}-\frac{42\!\cdots\!83}{52\!\cdots\!95}a^{12}+\frac{19\!\cdots\!36}{12\!\cdots\!25}a^{11}-\frac{41\!\cdots\!32}{27\!\cdots\!25}a^{10}+\frac{10\!\cdots\!58}{40\!\cdots\!75}a^{9}-\frac{35\!\cdots\!52}{12\!\cdots\!25}a^{8}+\frac{80\!\cdots\!34}{17\!\cdots\!75}a^{7}-\frac{34\!\cdots\!08}{24\!\cdots\!65}a^{6}+\frac{26\!\cdots\!06}{12\!\cdots\!25}a^{5}-\frac{81\!\cdots\!36}{40\!\cdots\!75}a^{4}+\frac{96\!\cdots\!49}{83\!\cdots\!75}a^{3}-\frac{19\!\cdots\!63}{45\!\cdots\!75}a^{2}+\frac{15\!\cdots\!07}{15\!\cdots\!25}a-\frac{94\!\cdots\!19}{88\!\cdots\!75}$, $\frac{72\!\cdots\!21}{12\!\cdots\!25}a^{27}-\frac{19\!\cdots\!97}{24\!\cdots\!65}a^{26}+\frac{10\!\cdots\!19}{24\!\cdots\!65}a^{25}-\frac{13\!\cdots\!87}{13\!\cdots\!25}a^{24}+\frac{37\!\cdots\!31}{24\!\cdots\!65}a^{23}-\frac{47\!\cdots\!08}{47\!\cdots\!25}a^{22}+\frac{71\!\cdots\!43}{13\!\cdots\!25}a^{21}-\frac{14\!\cdots\!09}{12\!\cdots\!25}a^{20}+\frac{12\!\cdots\!63}{12\!\cdots\!25}a^{19}-\frac{80\!\cdots\!49}{27\!\cdots\!85}a^{18}+\frac{14\!\cdots\!92}{79\!\cdots\!75}a^{17}-\frac{27\!\cdots\!47}{58\!\cdots\!25}a^{16}+\frac{87\!\cdots\!66}{14\!\cdots\!45}a^{15}-\frac{21\!\cdots\!58}{40\!\cdots\!75}a^{14}+\frac{45\!\cdots\!18}{40\!\cdots\!75}a^{13}-\frac{13\!\cdots\!82}{40\!\cdots\!75}a^{12}+\frac{47\!\cdots\!81}{81\!\cdots\!55}a^{11}-\frac{25\!\cdots\!98}{58\!\cdots\!25}a^{10}+\frac{39\!\cdots\!76}{12\!\cdots\!25}a^{9}-\frac{15\!\cdots\!49}{12\!\cdots\!25}a^{8}+\frac{73\!\cdots\!47}{34\!\cdots\!95}a^{7}-\frac{66\!\cdots\!53}{12\!\cdots\!25}a^{6}+\frac{91\!\cdots\!63}{12\!\cdots\!25}a^{5}-\frac{29\!\cdots\!37}{45\!\cdots\!75}a^{4}+\frac{62\!\cdots\!18}{17\!\cdots\!75}a^{3}-\frac{61\!\cdots\!29}{50\!\cdots\!75}a^{2}+\frac{16\!\cdots\!30}{60\!\cdots\!53}a-\frac{54\!\cdots\!44}{16\!\cdots\!25}$ 488611/510111886095a^26 - 6351943/510111886095a^25 + 13080549607/206595313868475a^24 - 9446583178/68865104622825a^23 + 38448811007/206595313868475a^22 - 62539442977/41319062773695a^21 + 557701827568/68865104622825a^20 - 194357120702/12152665521675a^19 + 1704112735522/206595313868475a^18 - 181908514049/4591006974855a^17 + 6619643874548/22955034874275a^16 - 140082159965218/206595313868475a^15 + 1631862790736/2430533104335a^14 - 85419870598606/206595313868475a^13 + 314590230468916/206595313868475a^12 - 29376148141217/5902723253385a^11 + 512019105626036/68865104622825a^10 - 627357685491028/206595313868475a^9 - 202792705471673/68865104622825a^8 - 4623655332811/1530335658285a^7 + 814389001367198/22955034874275a^6 - 15950274892656619/206595313868475a^5 + 1928957274236252/22955034874275a^4 - 309066381422812/5902723253385a^3 + 1468490012419/72873126585a^2 - 12501469099423/2550559430475a + 1630475588459/283395492275, 286781165271752606524439/90559206344971405172702400795a^27 - 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1907481638022442081009682197/2445098571314227939662964821465a^26 + 10049932644348006759787124219/2445098571314227939662964821465a^25 - 13296157610525265535125991987/1358388095174571077590536011925a^24 + 37849058773621415271145505231/2445098571314227939662964821465a^23 - 4730866107267169233879934508/47570011115062800382547953725a^22 + 711995133778872241918616933243/1358388095174571077590536011925a^21 - 14103009453487263774895140818609/12225492856571139698314824107325a^20 + 12252982522955749574248103567863/12225492856571139698314824107325a^19 - 804078596842035211965863015249/271677619034914215518107202385a^18 + 145421196965115267263812588292/7943790030260649576552842175a^17 - 27550752484559823461656800823447/582166326503387604681658290825a^16 + 8742020139319219979802541350566/143829327724366349391939107145a^15 - 216920592061503606980559349001758/4075164285523713232771608035775a^14 + 452386812607914213468086304814018/4075164285523713232771608035775a^13 - 1362182634855354420134517415788982/4075164285523713232771608035775a^12 + 472406416528592437118273674422481/815032857104742646554321607155a^11 - 256873152677025679980172773545098/582166326503387604681658290825a^10 + 391520154922946591832285956804476/12225492856571139698314824107325a^9 - 1564061860125259634928444957940349/12225492856571139698314824107325a^8 + 739638985332800250483150186791047/349299795902032562808994974495a^7 - 66433242307888659728038023905467453/12225492856571139698314824107325a^6 + 91474931635779672907678931185708463/12225492856571139698314824107325a^5 - 2925578287630999388670198750227837/452796031724857025863512003975a^4 + 6253347219713545498126803772531718/1746498979510162814044974872475a^3 - 61477324973268106112072870119229/50310670191650780651501333775a^2 + 1667743728610076313829941968930/6037280422998093678180160053*a - 548285137464353530839854719844/16770223397216926883833777925 (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:	$ 140953741391215.78 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{13}\cdot 140953741391215.78 \cdot 7}{2\cdot\sqrt{295815184798509371659078776791937755272938949172963}}\cr\approx \mathstrut & 2.72918327152723 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^28 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563)

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^28 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563, 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^28 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563);

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^28 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563);

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

$D_{28}$ (as 28T10):

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 56

The 17 conjugacy class representatives for $D_{28}$

Character table for $D_{28}$

Intermediate fields

$\Q(\sqrt{29}) $, 4.2.170723.1, 7.1.204024399103.1, 14.2.1207152707450866588933661.1

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 28 sibling:	deg 28
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	$28$	${\href{/padicField/3.2.0.1}{2} }^{14}$	${\href{/padicField/5.2.0.1}{2} }^{13}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$	R	$28$	${\href{/padicField/13.2.0.1}{2} }^{13}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.2.0.1}{2} }^{14}$	${\href{/padicField/19.2.0.1}{2} }^{14}$	${\href{/padicField/23.14.0.1}{14} }^{2}$	R	${\href{/padicField/31.2.0.1}{2} }^{14}$	$28$	${\href{/padicField/41.2.0.1}{2} }^{14}$	${\href{/padicField/43.4.0.1}{4} }^{7}$	${\href{/padicField/47.2.0.1}{2} }^{14}$	${\href{/padicField/53.7.0.1}{7} }^{4}$	${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$7$ 7	7.2.0.1	$x^{2} + 6 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	7.2.1.1	$x^{2} + 21$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$29$ 29		Deg $28$	$28$	$1$	$27$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
*	1.29.2t1.a.a	$1$	$ 29 $	$\Q(\sqrt{29}) $	$C_2$ (as 2T1)	$1$	$1$
	1.7.2t1.a.a	$1$	$ 7 $	$\Q(\sqrt{-7}) $	$C_2$ (as 2T1)	$1$	$-1$
	1.203.2t1.a.a	$1$	$ 7 \cdot 29 $	$\Q(\sqrt{-203}) $	$C_2$ (as 2T1)	$1$	$-1$
*	2.5887.4t3.c.a	$2$	$ 7 \cdot 29^{2}$	4.2.170723.1	$D_{4}$ (as 4T3)	$1$	$0$
*	2.5887.14t3.b.b	$2$	$ 7 \cdot 29^{2}$	14.2.1207152707450866588933661.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.5887.14t3.b.c	$2$	$ 7 \cdot 29^{2}$	14.2.1207152707450866588933661.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.5887.7t2.a.a	$2$	$ 7 \cdot 29^{2}$	7.1.204024399103.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.5887.14t3.b.a	$2$	$ 7 \cdot 29^{2}$	14.2.1207152707450866588933661.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.5887.7t2.a.b	$2$	$ 7 \cdot 29^{2}$	7.1.204024399103.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.5887.7t2.a.c	$2$	$ 7 \cdot 29^{2}$	7.1.204024399103.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.5887.28t10.a.f	$2$	$ 7 \cdot 29^{2}$	28.2.295815184798509371659078776791937755272938949172963.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.5887.28t10.a.b	$2$	$ 7 \cdot 29^{2}$	28.2.295815184798509371659078776791937755272938949172963.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.5887.28t10.a.a	$2$	$ 7 \cdot 29^{2}$	28.2.295815184798509371659078776791937755272938949172963.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.5887.28t10.a.d	$2$	$ 7 \cdot 29^{2}$	28.2.295815184798509371659078776791937755272938949172963.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.5887.28t10.a.c	$2$	$ 7 \cdot 29^{2}$	28.2.295815184798509371659078776791937755272938949172963.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.5887.28t10.a.e	$2$	$ 7 \cdot 29^{2}$	28.2.295815184798509371659078776791937755272938949172963.1	$D_{28}$ (as 28T10)	$1$	$0$

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 *.

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