Properties

Label 28.2.265...875.1
Degree $28$
Signature $[2, 13]$
Discriminant $-2.657\times 10^{45}$
Root discriminant \(41.91\)
Ramified primes $3,5,71$
Class number not computed
Class group not computed
Galois group $D_{28}$ (as 28T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059)
 
gp: K = bnfinit(y^28 - 9*y^27 + 53*y^26 - 226*y^25 + 695*y^24 - 1455*y^23 + 1352*y^22 + 4587*y^21 - 26437*y^20 + 75511*y^19 - 148243*y^18 + 166438*y^17 - 8850*y^16 - 553583*y^15 + 1599191*y^14 - 2514414*y^13 + 2904360*y^12 - 819447*y^11 - 1193138*y^10 + 5517125*y^9 - 17624126*y^8 + 7384001*y^7 - 17552991*y^6 + 11860646*y^5 - 8713168*y^4 - 7406098*y^3 - 9688274*y^2 - 1853761*y - 465059, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059)
 

\( x^{28} - 9 x^{27} + 53 x^{26} - 226 x^{25} + 695 x^{24} - 1455 x^{23} + 1352 x^{22} + 4587 x^{21} - 26437 x^{20} + 75511 x^{19} - 148243 x^{18} + 166438 x^{17} - 8850 x^{16} + \cdots - 465059 \) Copy content Toggle raw display

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:   \(-2657211910834657108824251865653514862060546875\) \(\medspace = -\,3^{14}\cdot 5^{21}\cdot 71^{13}\) 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:  \(41.91\)
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:  $3^{1/2}5^{3/4}71^{1/2}\approx 48.79971717169349$
Ramified primes:   \(3\), \(5\), \(71\) 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:  \(\Q(\sqrt{-355}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7}a^{18}-\frac{1}{7}a^{17}-\frac{3}{7}a^{14}+\frac{1}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{10}+\frac{2}{7}a^{9}+\frac{2}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{19}-\frac{1}{7}a^{17}-\frac{3}{7}a^{15}-\frac{2}{7}a^{14}+\frac{2}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}-\frac{3}{7}a^{10}+\frac{2}{7}a^{9}+\frac{2}{7}a^{8}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}-\frac{1}{7}a^{4}+\frac{3}{7}a^{3}+\frac{3}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{20}-\frac{1}{7}a^{17}-\frac{3}{7}a^{16}-\frac{2}{7}a^{15}-\frac{1}{7}a^{14}+\frac{2}{7}a^{13}+\frac{3}{7}a^{12}-\frac{3}{7}a^{11}-\frac{3}{7}a^{10}-\frac{3}{7}a^{9}+\frac{3}{7}a^{6}-\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{3}{7}a^{3}+\frac{3}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{21}+\frac{3}{7}a^{17}-\frac{2}{7}a^{16}-\frac{1}{7}a^{15}-\frac{1}{7}a^{14}-\frac{3}{7}a^{13}-\frac{2}{7}a^{12}-\frac{3}{7}a^{11}-\frac{1}{7}a^{10}+\frac{2}{7}a^{9}-\frac{2}{7}a^{7}-\frac{3}{7}a^{6}+\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{3}{7}a^{3}+\frac{3}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{22}+\frac{1}{7}a^{17}-\frac{1}{7}a^{16}-\frac{1}{7}a^{15}-\frac{1}{7}a^{14}+\frac{2}{7}a^{13}+\frac{1}{7}a^{12}-\frac{1}{7}a^{11}+\frac{3}{7}a^{10}+\frac{1}{7}a^{9}-\frac{2}{7}a^{8}-\frac{2}{7}a^{7}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}a^{2}-\frac{3}{7}a$, $\frac{1}{91}a^{23}+\frac{2}{91}a^{21}-\frac{4}{91}a^{20}+\frac{4}{91}a^{19}-\frac{6}{91}a^{18}-\frac{9}{91}a^{17}-\frac{5}{13}a^{16}+\frac{5}{13}a^{15}-\frac{32}{91}a^{14}+\frac{2}{91}a^{13}-\frac{34}{91}a^{12}+\frac{17}{91}a^{11}+\frac{1}{7}a^{10}+\frac{22}{91}a^{9}+\frac{27}{91}a^{8}-\frac{18}{91}a^{7}+\frac{25}{91}a^{6}-\frac{27}{91}a^{5}-\frac{3}{7}a^{4}+\frac{20}{91}a^{3}-\frac{4}{91}a^{2}-\frac{5}{91}a-\frac{4}{13}$, $\frac{1}{71071}a^{24}-\frac{290}{71071}a^{23}-\frac{334}{10153}a^{22}-\frac{584}{71071}a^{21}+\frac{1944}{71071}a^{20}-\frac{106}{6461}a^{19}+\frac{2004}{71071}a^{18}+\frac{32397}{71071}a^{17}+\frac{4523}{10153}a^{16}-\frac{27875}{71071}a^{15}-\frac{1217}{5467}a^{14}+\frac{18561}{71071}a^{13}-\frac{4491}{10153}a^{12}-\frac{33673}{71071}a^{11}-\frac{3098}{71071}a^{10}+\frac{32920}{71071}a^{9}+\frac{12484}{71071}a^{8}-\frac{18012}{71071}a^{7}+\frac{11716}{71071}a^{6}-\frac{4152}{10153}a^{5}-\frac{17660}{71071}a^{4}+\frac{3123}{6461}a^{3}-\frac{1390}{6461}a^{2}-\frac{29908}{71071}a-\frac{4157}{10153}$, $\frac{1}{71071}a^{25}+\frac{23}{6461}a^{23}+\frac{1647}{71071}a^{22}-\frac{4187}{71071}a^{21}+\frac{2617}{71071}a^{20}+\frac{475}{71071}a^{19}+\frac{2034}{71071}a^{18}+\frac{1502}{6461}a^{17}-\frac{166}{923}a^{16}-\frac{19345}{71071}a^{15}+\frac{27437}{71071}a^{14}-\frac{18903}{71071}a^{13}+\frac{14632}{71071}a^{12}-\frac{9673}{71071}a^{11}+\frac{27964}{71071}a^{10}-\frac{16557}{71071}a^{9}+\frac{13653}{71071}a^{8}+\frac{20155}{71071}a^{7}+\frac{12619}{71071}a^{6}-\frac{34850}{71071}a^{5}-\frac{394}{71071}a^{4}+\frac{450}{6461}a^{3}+\frac{250}{10153}a^{2}+\frac{1089}{6461}a+\frac{1115}{10153}$, $\frac{1}{600478879}a^{26}+\frac{809}{600478879}a^{25}+\frac{729}{600478879}a^{24}+\frac{869590}{600478879}a^{23}+\frac{4438996}{600478879}a^{22}-\frac{957137}{85782697}a^{21}-\frac{32895524}{600478879}a^{20}-\frac{3422414}{54588989}a^{19}+\frac{284860}{4199153}a^{18}+\frac{154790400}{600478879}a^{17}+\frac{8899335}{85782697}a^{16}-\frac{14399713}{600478879}a^{15}+\frac{20378690}{46190683}a^{14}+\frac{198118223}{600478879}a^{13}+\frac{12620827}{54588989}a^{12}+\frac{297909452}{600478879}a^{11}-\frac{10554207}{54588989}a^{10}-\frac{144489837}{600478879}a^{9}+\frac{108597248}{600478879}a^{8}-\frac{71089838}{600478879}a^{7}+\frac{85248803}{600478879}a^{6}-\frac{150015035}{600478879}a^{5}-\frac{13562937}{600478879}a^{4}-\frac{97919436}{600478879}a^{3}-\frac{282181388}{600478879}a^{2}+\frac{3041856}{12254671}a-\frac{5485143}{12254671}$, $\frac{1}{70\!\cdots\!27}a^{27}+\frac{53\!\cdots\!61}{10\!\cdots\!61}a^{26}-\frac{41\!\cdots\!09}{64\!\cdots\!57}a^{25}-\frac{30\!\cdots\!09}{70\!\cdots\!27}a^{24}-\frac{36\!\cdots\!09}{70\!\cdots\!27}a^{23}+\frac{33\!\cdots\!37}{27\!\cdots\!61}a^{22}-\frac{28\!\cdots\!03}{70\!\cdots\!27}a^{21}-\frac{34\!\cdots\!55}{70\!\cdots\!27}a^{20}+\frac{93\!\cdots\!60}{70\!\cdots\!27}a^{19}-\frac{68\!\cdots\!40}{54\!\cdots\!79}a^{18}-\frac{39\!\cdots\!26}{70\!\cdots\!27}a^{17}-\frac{19\!\cdots\!11}{70\!\cdots\!27}a^{16}-\frac{53\!\cdots\!30}{67\!\cdots\!89}a^{15}+\frac{23\!\cdots\!02}{11\!\cdots\!71}a^{14}-\frac{19\!\cdots\!44}{10\!\cdots\!61}a^{13}-\frac{23\!\cdots\!27}{70\!\cdots\!27}a^{12}+\frac{16\!\cdots\!38}{77\!\cdots\!97}a^{11}+\frac{22\!\cdots\!95}{70\!\cdots\!27}a^{10}-\frac{11\!\cdots\!12}{24\!\cdots\!57}a^{9}+\frac{13\!\cdots\!92}{70\!\cdots\!27}a^{8}-\frac{10\!\cdots\!98}{10\!\cdots\!61}a^{7}-\frac{28\!\cdots\!94}{10\!\cdots\!61}a^{6}-\frac{29\!\cdots\!22}{64\!\cdots\!57}a^{5}+\frac{39\!\cdots\!91}{70\!\cdots\!27}a^{4}-\frac{17\!\cdots\!44}{70\!\cdots\!27}a^{3}-\frac{12\!\cdots\!10}{70\!\cdots\!27}a^{2}+\frac{78\!\cdots\!07}{59\!\cdots\!33}a-\frac{25\!\cdots\!17}{14\!\cdots\!23}$ Copy content Toggle raw display

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

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

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

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059)
 
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 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059, 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 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059);
 
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 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059);
 
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

\(\Q(\sqrt{5}) \), 4.2.79875.1, 7.1.357911.1, 14.2.10007834681328125.1

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 $28$ R R ${\href{/padicField/7.2.0.1}{2} }^{14}$ ${\href{/padicField/11.2.0.1}{2} }^{13}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{14}$ ${\href{/padicField/17.2.0.1}{2} }^{14}$ ${\href{/padicField/19.7.0.1}{7} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{14}$ ${\href{/padicField/29.14.0.1}{14} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{13}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $28$ ${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $28$ ${\href{/padicField/47.2.0.1}{2} }^{14}$ ${\href{/padicField/53.2.0.1}{2} }^{14}$ ${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $28$$2$$14$$14$
\(5\) Copy content Toggle raw display Deg $28$$4$$7$$21$
\(71\) Copy content Toggle raw display $\Q_{71}$$x + 64$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 64$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.355.2t1.a.a$1$ $ 5 \cdot 71 $ \(\Q(\sqrt{-355}) \) $C_2$ (as 2T1) $1$ $-1$
1.71.2t1.a.a$1$ $ 71 $ \(\Q(\sqrt{-71}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.15975.4t3.a.a$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 4.0.5671125.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.1775.14t3.a.c$2$ $ 5^{2} \cdot 71 $ 14.0.710556262374296875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.1775.14t3.a.b$2$ $ 5^{2} \cdot 71 $ 14.0.710556262374296875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.c$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.a.b$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.1775.14t3.a.a$2$ $ 5^{2} \cdot 71 $ 14.0.710556262374296875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.a$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.15975.28t10.a.d$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.15975.28t10.a.c$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.15975.28t10.a.e$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.15975.28t10.a.a$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.15975.28t10.a.b$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.15975.28t10.a.f$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (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 *.