Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 44*x^18 + 770*x^16 - 6952*x^14 + 35552*x^12 - 107712*x^10 + 196504*x^8 - 212960*x^6 + 129712*x^4 - 38720*x^2 + 3872)
gp: K = bnfinit(y^20 - 44*y^18 + 770*y^16 - 6952*y^14 + 35552*y^12 - 107712*y^10 + 196504*y^8 - 212960*y^6 + 129712*y^4 - 38720*y^2 + 3872, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 44*x^18 + 770*x^16 - 6952*x^14 + 35552*x^12 - 107712*x^10 + 196504*x^8 - 212960*x^6 + 129712*x^4 - 38720*x^2 + 3872);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 44*x^18 + 770*x^16 - 6952*x^14 + 35552*x^12 - 107712*x^10 + 196504*x^8 - 212960*x^6 + 129712*x^4 - 38720*x^2 + 3872)
\( x^{20} - 44 x^{18} + 770 x^{16} - 6952 x^{14} + 35552 x^{12} - 107712 x^{10} + 196504 x^{8} + \cdots + 3872 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[20, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(200317132330035063121671003054276608\)
\(\medspace = 2^{55}\cdot 11^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(58.22\) | 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: | $2^{11/4}11^{9/10}\approx 58.22183708777889$ | ||
| Ramified primes: |
\(2\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(176=2^{4}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{176}(1,·)$, $\chi_{176}(131,·)$, $\chi_{176}(81,·)$, $\chi_{176}(9,·)$, $\chi_{176}(139,·)$, $\chi_{176}(43,·)$, $\chi_{176}(83,·)$, $\chi_{176}(171,·)$, $\chi_{176}(25,·)$, $\chi_{176}(89,·)$, $\chi_{176}(51,·)$, $\chi_{176}(97,·)$, $\chi_{176}(35,·)$, $\chi_{176}(113,·)$, $\chi_{176}(169,·)$, $\chi_{176}(107,·)$, $\chi_{176}(49,·)$, $\chi_{176}(19,·)$, $\chi_{176}(137,·)$, $\chi_{176}(123,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{44}a^{10}$, $\frac{1}{44}a^{11}$, $\frac{1}{88}a^{12}$, $\frac{1}{88}a^{13}$, $\frac{1}{88}a^{14}$, $\frac{1}{88}a^{15}$, $\frac{1}{4048}a^{16}-\frac{1}{2024}a^{14}+\frac{5}{2024}a^{12}-\frac{7}{1012}a^{10}-\frac{2}{23}a^{8}-\frac{3}{23}a^{6}-\frac{7}{46}a^{4}+\frac{4}{23}a^{2}+\frac{5}{23}$, $\frac{1}{4048}a^{17}-\frac{1}{2024}a^{15}+\frac{5}{2024}a^{13}-\frac{7}{1012}a^{11}-\frac{2}{23}a^{9}-\frac{3}{23}a^{7}-\frac{7}{46}a^{5}+\frac{4}{23}a^{3}+\frac{5}{23}a$, $\frac{1}{805552}a^{18}+\frac{45}{805552}a^{16}+\frac{945}{201388}a^{14}+\frac{87}{36616}a^{12}-\frac{340}{50347}a^{10}-\frac{396}{4577}a^{8}-\frac{197}{9154}a^{6}-\frac{126}{4577}a^{4}-\frac{129}{4577}a^{2}-\frac{1950}{4577}$, $\frac{1}{805552}a^{19}+\frac{45}{805552}a^{17}+\frac{945}{201388}a^{15}+\frac{87}{36616}a^{13}-\frac{340}{50347}a^{11}-\frac{396}{4577}a^{9}-\frac{197}{9154}a^{7}-\frac{126}{4577}a^{5}-\frac{129}{4577}a^{3}-\frac{1950}{4577}a$
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: | $19$ | 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: |
$\frac{67}{50347}a^{18}-\frac{51061}{805552}a^{16}+\frac{5584}{4577}a^{14}-\frac{4935539}{402776}a^{12}+\frac{635025}{9154}a^{10}-\frac{2049193}{9154}a^{8}+\frac{1830877}{4577}a^{6}-\frac{3383101}{9154}a^{4}+\frac{700298}{4577}a^{2}-\frac{87863}{4577}$, $\frac{1515}{402776}a^{18}-\frac{126529}{805552}a^{16}+\frac{511767}{201388}a^{14}-\frac{2046645}{100694}a^{12}+\frac{787809}{9154}a^{10}-\frac{3570695}{18308}a^{8}+\frac{1039422}{4577}a^{6}-\frac{1074597}{9154}a^{4}+\frac{77178}{4577}a^{2}-\frac{4966}{4577}$, $\frac{868}{50347}a^{18}-\frac{36958}{50347}a^{16}+\frac{1231865}{100694}a^{14}-\frac{5163353}{50347}a^{12}+\frac{2144691}{4577}a^{10}-\frac{22003567}{18308}a^{8}+\frac{7883144}{4577}a^{6}-\frac{5996946}{4577}a^{4}+\frac{2112828}{4577}a^{2}-\frac{231525}{4577}$, $\frac{117}{17512}a^{18}-\frac{230833}{805552}a^{16}+\frac{972063}{201388}a^{14}-\frac{16556943}{402776}a^{12}+\frac{879537}{4577}a^{10}-\frac{9304173}{18308}a^{8}+\frac{3456639}{4577}a^{6}-\frac{2727633}{4577}a^{4}+\frac{992844}{4577}a^{2}-\frac{117647}{4577}$, $\frac{9863}{805552}a^{18}-\frac{52901}{100694}a^{16}+\frac{1783211}{201388}a^{14}-\frac{3799442}{50347}a^{12}+\frac{71105401}{201388}a^{10}-\frac{4277082}{4577}a^{8}+\frac{6359953}{4577}a^{6}-\frac{5020952}{4577}a^{4}+\frac{1813765}{4577}a^{2}-\frac{188749}{4577}$, $\frac{4863}{805552}a^{18}-\frac{209413}{805552}a^{16}+\frac{443585}{100694}a^{14}-\frac{693213}{18308}a^{12}+\frac{18079093}{100694}a^{10}-\frac{8886047}{18308}a^{8}+\frac{3416917}{4577}a^{6}-\frac{2853279}{4577}a^{4}+\frac{49487}{199}a^{2}-\frac{149551}{4577}$, $\frac{1013}{73232}a^{18}-\frac{476451}{805552}a^{16}+\frac{1997811}{201388}a^{14}-\frac{4228795}{50347}a^{12}+\frac{78502681}{201388}a^{10}-\frac{4685855}{4577}a^{8}+\frac{6946293}{4577}a^{6}-\frac{5519736}{4577}a^{4}+\frac{2029133}{4577}a^{2}-\frac{218144}{4577}$, $\frac{175}{35024}a^{18}-\frac{21015}{100694}a^{16}+\frac{680519}{201388}a^{14}-\frac{1363911}{50347}a^{12}+\frac{23261003}{201388}a^{10}-\frac{4895239}{18308}a^{8}+\frac{1523191}{4577}a^{6}-\frac{975994}{4577}a^{4}+\frac{299063}{4577}a^{2}-\frac{42776}{4577}$, $\frac{315}{805552}a^{18}-\frac{10501}{805552}a^{16}+\frac{26239}{201388}a^{14}-\frac{14159}{402776}a^{12}-\frac{28841}{4378}a^{10}+\frac{176347}{4577}a^{8}-\frac{801937}{9154}a^{6}+\frac{793633}{9154}a^{4}-\frac{148493}{4577}a^{2}+\frac{8421}{4577}$, $\frac{85}{9154}a^{19}-\frac{175}{35024}a^{18}-\frac{19857}{50347}a^{17}+\frac{21015}{100694}a^{16}+\frac{2639577}{402776}a^{15}-\frac{680519}{201388}a^{14}-\frac{22060213}{402776}a^{13}+\frac{1363911}{50347}a^{12}+\frac{25173739}{100694}a^{11}-\frac{23261003}{201388}a^{10}-\frac{11827027}{18308}a^{9}+\frac{4895239}{18308}a^{8}+\frac{8727379}{9154}a^{7}-\frac{1523191}{4577}a^{6}-\frac{7149903}{9154}a^{5}+\frac{975994}{4577}a^{4}+\frac{1461686}{4577}a^{3}-\frac{299063}{4577}a^{2}-\frac{208648}{4577}a+\frac{47353}{4577}$, $\frac{135}{18308}a^{19}-\frac{17597}{805552}a^{18}-\frac{250697}{805552}a^{17}+\frac{758345}{805552}a^{16}+\frac{1032121}{201388}a^{15}-\frac{1607785}{100694}a^{14}-\frac{4259907}{100694}a^{13}+\frac{27661421}{201388}a^{12}+\frac{9570757}{50347}a^{11}-\frac{131172859}{201388}a^{10}-\frac{2213338}{4577}a^{9}+\frac{32145979}{18308}a^{8}+\frac{6473609}{9154}a^{7}-\frac{12233809}{4577}a^{6}-\frac{2640532}{4577}a^{5}+\frac{19877839}{9154}a^{4}+\frac{1038471}{4577}a^{3}-\frac{3722275}{4577}a^{2}-\frac{117018}{4577}a+\frac{419077}{4577}$, $\frac{1049}{402776}a^{19}-\frac{39}{36616}a^{18}-\frac{8875}{73232}a^{17}+\frac{10745}{201388}a^{16}+\frac{20939}{9154}a^{15}-\frac{219979}{201388}a^{14}-\frac{9195715}{402776}a^{13}+\frac{596001}{50347}a^{12}+\frac{13335777}{100694}a^{11}-\frac{3711762}{50347}a^{10}-\frac{8483103}{18308}a^{9}+\frac{1233875}{4577}a^{8}+\frac{8954619}{9154}a^{7}-\frac{5249943}{9154}a^{6}-\frac{10993111}{9154}a^{5}+\frac{6258851}{9154}a^{4}+\frac{3556526}{4577}a^{3}-\frac{1897228}{4577}a^{2}-\frac{922821}{4577}a+\frac{442484}{4577}$, $\frac{47}{3184}a^{19}+\frac{9369}{805552}a^{18}-\frac{510451}{805552}a^{17}-\frac{49859}{100694}a^{16}+\frac{195695}{18308}a^{15}+\frac{3323093}{402776}a^{14}-\frac{18378709}{201388}a^{13}-\frac{27848071}{402776}a^{12}+\frac{21580300}{50347}a^{11}+\frac{63587993}{201388}a^{10}-\frac{10476153}{9154}a^{9}-\frac{7410895}{9154}a^{8}+\frac{15895027}{9154}a^{7}+\frac{10638373}{9154}a^{6}-\frac{13148119}{9154}a^{5}-\frac{8188305}{9154}a^{4}+\frac{2655210}{4577}a^{3}+\frac{1519092}{4577}a^{2}-\frac{385632}{4577}a-\frac{205524}{4577}$, $\frac{35}{18308}a^{19}-\frac{7511}{805552}a^{18}-\frac{67015}{805552}a^{17}+\frac{39863}{100694}a^{16}+\frac{575335}{402776}a^{15}-\frac{120265}{18308}a^{14}-\frac{5020585}{402776}a^{13}+\frac{22025173}{402776}a^{12}+\frac{6032225}{100694}a^{11}-\frac{49737571}{201388}a^{10}-\frac{2973675}{18308}a^{9}+\frac{5687425}{9154}a^{8}+\frac{48995}{199}a^{7}-\frac{3942736}{4577}a^{6}-\frac{1868839}{9154}a^{5}+\frac{5661235}{9154}a^{4}+\frac{423215}{4577}a^{3}-\frac{898099}{4577}a^{2}-\frac{91630}{4577}a+\frac{71491}{4577}$, $\frac{4481}{805552}a^{18}-\frac{192375}{805552}a^{16}+\frac{202787}{50347}a^{14}-\frac{13838593}{402776}a^{12}+\frac{32405773}{201388}a^{10}-\frac{7804155}{18308}a^{8}+\frac{2903314}{4577}a^{6}-\frac{2293319}{4577}a^{4}+\frac{820921}{4577}a^{2}-a-\frac{80256}{4577}$, $\frac{917}{100694}a^{19}-\frac{2421}{201388}a^{18}-\frac{310859}{805552}a^{17}+\frac{102791}{201388}a^{16}+\frac{58495}{9154}a^{15}-\frac{1706989}{201388}a^{14}-\frac{1945783}{36616}a^{13}+\frac{7126847}{100694}a^{12}+\frac{24261535}{100694}a^{11}-\frac{16245857}{50347}a^{10}-\frac{11321127}{18308}a^{9}+\frac{3811581}{4577}a^{8}+\frac{8355359}{9154}a^{7}-\frac{11252321}{9154}a^{6}-\frac{7046751}{9154}a^{5}+\frac{9267859}{9154}a^{4}+\frac{70658}{199}a^{3}-\frac{1926406}{4577}a^{2}-\frac{346345}{4577}a+\frac{288444}{4577}$, $\frac{8665}{805552}a^{19}+\frac{109}{17512}a^{18}-\frac{93111}{201388}a^{17}-\frac{108473}{402776}a^{16}+\frac{1573665}{201388}a^{15}+\frac{1849785}{402776}a^{14}-\frac{26949699}{402776}a^{13}-\frac{4006141}{100694}a^{12}+\frac{15891638}{50347}a^{11}+\frac{38300429}{201388}a^{10}-\frac{15521733}{18308}a^{9}-\frac{4722711}{9154}a^{8}+\frac{11847283}{9154}a^{7}+\frac{3593780}{4577}a^{6}-\frac{4893088}{4577}a^{5}-\frac{5767893}{9154}a^{4}+\frac{1901841}{4577}a^{3}+\frac{1047679}{4577}a^{2}-\frac{231097}{4577}a-\frac{112865}{4577}$, $\frac{8161}{805552}a^{19}-\frac{23329}{805552}a^{18}-\frac{7569}{18308}a^{17}+\frac{977607}{805552}a^{16}+\frac{324942}{50347}a^{15}-\frac{1989049}{100694}a^{14}-\frac{4904971}{100694}a^{13}+\frac{16090383}{100694}a^{12}+\frac{18832199}{100694}a^{11}-\frac{69714041}{100694}a^{10}-\frac{1638960}{4577}a^{9}+\frac{30194499}{18308}a^{8}+\frac{1390813}{4577}a^{7}-\frac{9860297}{4577}a^{6}-\frac{259350}{4577}a^{5}+\frac{6755080}{4577}a^{4}-\frac{210601}{4577}a^{3}-\frac{2117595}{4577}a^{2}+\frac{59780}{4577}a+\frac{205717}{4577}$, $\frac{147}{50347}a^{19}+\frac{285}{805552}a^{18}-\frac{6519}{50347}a^{17}-\frac{2913}{201388}a^{16}+\frac{115074}{50347}a^{15}+\frac{1042}{4577}a^{14}-\frac{8370363}{402776}a^{13}-\frac{16139}{9154}a^{12}+\frac{971265}{9154}a^{11}+\frac{1468075}{201388}a^{10}-\frac{2866739}{9154}a^{9}-\frac{77637}{4577}a^{8}+\frac{2417217}{4577}a^{7}+\frac{102571}{4577}a^{6}-\frac{4380669}{9154}a^{5}-\frac{120177}{9154}a^{4}+\frac{915666}{4577}a^{3}-\frac{6517}{4577}a^{2}-\frac{117258}{4577}a+\frac{8415}{4577}$
| 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: | \( 228779211109 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 228779211109 \cdot 1}{2\cdot\sqrt{200317132330035063121671003054276608}}\cr\approx \mathstrut & 0.267995454842375 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 44*x^18 + 770*x^16 - 6952*x^14 + 35552*x^12 - 107712*x^10 + 196504*x^8 - 212960*x^6 + 129712*x^4 - 38720*x^2 + 3872)
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^20 - 44*x^18 + 770*x^16 - 6952*x^14 + 35552*x^12 - 107712*x^10 + 196504*x^8 - 212960*x^6 + 129712*x^4 - 38720*x^2 + 3872, 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^20 - 44*x^18 + 770*x^16 - 6952*x^14 + 35552*x^12 - 107712*x^10 + 196504*x^8 - 212960*x^6 + 129712*x^4 - 38720*x^2 + 3872);
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^20 - 44*x^18 + 770*x^16 - 6952*x^14 + 35552*x^12 - 107712*x^10 + 196504*x^8 - 212960*x^6 + 129712*x^4 - 38720*x^2 + 3872);
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 cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.247808.1, \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $20$ | $20$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | $20$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/23.1.0.1}{1} }^{20}$ | $20$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | $20$ | $20$ |
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 $20$ | $4$ | $5$ | $55$ | |||
|
\(11\)
| 11.20.18.8 | $x^{20} - 704 x^{10} - 121$ | $10$ | $2$ | $18$ | 20T1 | $[\ ]_{10}^{2}$ |