LMFDB - Number field 16.12.121821863230510940507078656.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 16*x^12 + 236*x^10 - 638*x^8 - 908*x^6 + 2944*x^4 - 756*x^2 + 49)

gp:K = bnfinit(y^16 - 12*y^14 + 16*y^12 + 236*y^10 - 638*y^8 - 908*y^6 + 2944*y^4 - 756*y^2 + 49, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 16*x^12 + 236*x^10 - 638*x^8 - 908*x^6 + 2944*x^4 - 756*x^2 + 49);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 12*x^14 + 16*x^12 + 236*x^10 - 638*x^8 - 908*x^6 + 2944*x^4 - 756*x^2 + 49)

$ x^{16} - 12x^{14} + 16x^{12} + 236x^{10} - 638x^{8} - 908x^{6} + 2944x^{4} - 756x^{2} + 49 $ x^16 - 12*x^14 + 16*x^12 + 236*x^10 - 638*x^8 - 908*x^6 + 2944*x^4 - 756*x^2 + 49

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$16$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[12, 2]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$121821863230510940507078656$ 121821863230510940507078656 $\medspace = 2^{50}\cdot 7^{8}\cdot 137^{2}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$42.69$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	$2^{115/32}7^{1/2}137^{1/2}\approx 373.8831134106666$
Ramified primes:	$2$, $7$, $137$ 2, 7, 137	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant(OK))
Discriminant root field:	$\Q$
$\Aut(K/\Q)$:	$C_2$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}+\frac{1}{4}$, $\frac{1}{4}a^{9}+\frac{1}{4}a$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{3}$, $\frac{1}{12}a^{12}-\frac{1}{12}a^{10}+\frac{1}{3}a^{6}-\frac{1}{4}a^{4}-\frac{1}{12}a^{2}-\frac{1}{3}$, $\frac{1}{24}a^{13}-\frac{1}{24}a^{12}+\frac{1}{12}a^{11}-\frac{1}{12}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{5}{12}a^{3}+\frac{5}{12}a^{2}-\frac{7}{24}a+\frac{1}{24}$, $\frac{1}{591864}a^{14}-\frac{12059}{591864}a^{12}+\frac{21499}{591864}a^{10}+\frac{40927}{591864}a^{8}-\frac{25495}{591864}a^{6}+\frac{157229}{591864}a^{4}-\frac{7351}{197288}a^{2}-\frac{38927}{84552}$, $\frac{1}{4143048}a^{15}+\frac{6301}{2071524}a^{13}-\frac{1}{24}a^{12}-\frac{124359}{1381016}a^{11}-\frac{1}{12}a^{10}-\frac{90511}{2071524}a^{9}-\frac{1}{8}a^{8}-\frac{370193}{1381016}a^{7}-\frac{1}{6}a^{6}-\frac{254309}{2071524}a^{5}+\frac{1}{8}a^{4}-\frac{712561}{4143048}a^{3}+\frac{5}{12}a^{2}-\frac{14121}{98644}a+\frac{1}{24}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/4*a^8 + 1/4, 1/4*a^9 + 1/4*a, 1/4*a^10 + 1/4*a^2, 1/4*a^11 + 1/4*a^3, 1/12*a^12 - 1/12*a^10 + 1/3*a^6 - 1/4*a^4 - 1/12*a^2 - 1/3, 1/24*a^13 - 1/24*a^12 + 1/12*a^11 - 1/12*a^10 - 1/8*a^9 - 1/8*a^8 + 1/6*a^7 - 1/6*a^6 - 1/8*a^5 + 1/8*a^4 - 5/12*a^3 + 5/12*a^2 - 7/24*a + 1/24, 1/591864*a^14 - 12059/591864*a^12 + 21499/591864*a^10 + 40927/591864*a^8 - 25495/591864*a^6 + 157229/591864*a^4 - 7351/197288*a^2 - 38927/84552, 1/4143048*a^15 + 6301/2071524*a^13 - 1/24*a^12 - 124359/1381016*a^11 - 1/12*a^10 - 90511/2071524*a^9 - 1/8*a^8 - 370193/1381016*a^7 - 1/6*a^6 - 254309/2071524*a^5 + 1/8*a^4 - 712561/4143048*a^3 + 5/12*a^2 - 14121/98644*a + 1/24

comment:Integral basis

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

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$13$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity 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:	$\frac{99}{14092}a^{14}-\frac{2155}{42276}a^{12}-\frac{5519}{42276}a^{10}+\frac{14415}{14092}a^{8}+\frac{23573}{42276}a^{6}-\frac{48265}{14092}a^{4}+\frac{38269}{42276}a^{2}-\frac{20015}{42276}$, $\frac{2677}{591864}a^{14}-\frac{25355}{591864}a^{12}-\frac{5951}{591864}a^{10}+\frac{510637}{591864}a^{8}-\frac{777619}{591864}a^{6}-\frac{1984795}{591864}a^{4}+\frac{1776443}{197288}a^{2}-\frac{60653}{84552}$, $\frac{20}{517881}a^{15}-\frac{13801}{1035762}a^{13}+\frac{31823}{345254}a^{11}+\frac{536789}{2071524}a^{9}-\frac{326153}{172627}a^{7}-\frac{1183123}{1035762}a^{5}+\frac{7231349}{1035762}a^{3}-\frac{261819}{98644}a$, $\frac{103009}{2071524}a^{15}-\frac{1244011}{2071524}a^{13}+\frac{553439}{690508}a^{11}+\frac{6196594}{517881}a^{9}-\frac{22002693}{690508}a^{7}-\frac{101269111}{2071524}a^{5}+\frac{303592109}{2071524}a^{3}-\frac{572366}{24661}a$, $\frac{32503}{4143048}a^{15}-\frac{401}{73983}a^{14}-\frac{53423}{1035762}a^{13}+\frac{41485}{591864}a^{12}-\frac{801251}{4143048}a^{11}-\frac{10993}{98644}a^{10}+\frac{568613}{517881}a^{9}-\frac{861787}{591864}a^{8}+\frac{6752051}{4143048}a^{7}+\frac{198295}{49322}a^{6}-\frac{2307161}{517881}a^{5}+\frac{3946561}{591864}a^{4}-\frac{4847341}{1381016}a^{3}-\frac{5490265}{295932}a^{2}+\frac{159139}{147966}a+\frac{69805}{28184}$, $\frac{32503}{4143048}a^{15}+\frac{401}{73983}a^{14}-\frac{53423}{1035762}a^{13}-\frac{41485}{591864}a^{12}-\frac{801251}{4143048}a^{11}+\frac{10993}{98644}a^{10}+\frac{568613}{517881}a^{9}+\frac{861787}{591864}a^{8}+\frac{6752051}{4143048}a^{7}-\frac{198295}{49322}a^{6}-\frac{2307161}{517881}a^{5}-\frac{3946561}{591864}a^{4}-\frac{4847341}{1381016}a^{3}+\frac{5490265}{295932}a^{2}+\frac{159139}{147966}a-\frac{69805}{28184}$, $\frac{38023}{1381016}a^{15}+\frac{7127}{591864}a^{14}-\frac{1348507}{4143048}a^{13}-\frac{74891}{591864}a^{12}+\frac{1559251}{4143048}a^{11}+\frac{9747}{197288}a^{10}+\frac{9133469}{1381016}a^{9}+\frac{1525403}{591864}a^{8}-\frac{67636531}{4143048}a^{7}-\frac{920883}{197288}a^{6}-\frac{39656241}{1381016}a^{5}-\frac{6485939}{591864}a^{4}+\frac{314828635}{4143048}a^{3}+\frac{13383745}{591864}a^{2}-\frac{3145463}{591864}a-\frac{22313}{28184}$, $\frac{4603}{147966}a^{14}-\frac{7498}{24661}a^{12}-\frac{2348}{73983}a^{10}+\frac{914059}{147966}a^{8}-\frac{1298819}{147966}a^{6}-\frac{1874876}{73983}a^{4}+\frac{3486890}{73983}a^{2}-\frac{106891}{21138}$, $\frac{1397}{98644}a^{14}-\frac{33541}{295932}a^{12}-\frac{58307}{295932}a^{10}+\frac{116385}{49322}a^{8}-\frac{116737}{295932}a^{6}-\frac{919071}{98644}a^{4}+\frac{2372761}{295932}a^{2}-\frac{19363}{21138}$, $\frac{150145}{4143048}a^{15}+\frac{2001}{197288}a^{14}-\frac{132575}{345254}a^{13}-\frac{5071}{295932}a^{12}+\frac{779573}{4143048}a^{11}-\frac{362431}{591864}a^{10}+\frac{15995537}{2071524}a^{9}+\frac{47195}{98644}a^{8}-\frac{62014787}{4143048}a^{7}+\frac{5671711}{591864}a^{6}-\frac{16148450}{517881}a^{5}-\frac{115879}{98644}a^{4}+\frac{307459489}{4143048}a^{3}-\frac{19436779}{591864}a^{2}-\frac{3874631}{295932}a+\frac{231661}{42276}$, $\frac{275}{39837}a^{15}-\frac{24727}{591864}a^{14}-\frac{13817}{79674}a^{13}+\frac{179369}{591864}a^{12}+\frac{61153}{79674}a^{11}+\frac{479963}{591864}a^{10}+\frac{134711}{39837}a^{9}-\frac{3761605}{591864}a^{8}-\frac{722570}{39837}a^{7}-\frac{2287751}{591864}a^{6}-\frac{1159759}{79674}a^{5}+\frac{15245665}{591864}a^{4}+\frac{1998895}{26558}a^{3}-\frac{723223}{197288}a^{2}-\frac{76576}{5691}a-\frac{118867}{84552}$, $\frac{12829}{1381016}a^{15}-\frac{4961}{295932}a^{14}-\frac{586577}{4143048}a^{13}+\frac{3629}{49322}a^{12}+\frac{1535123}{4143048}a^{11}+\frac{99787}{147966}a^{10}+\frac{3817587}{1381016}a^{9}-\frac{473393}{295932}a^{8}-\frac{43910777}{4143048}a^{7}-\frac{2644301}{295932}a^{6}-\frac{15468779}{1381016}a^{5}+\frac{808663}{147966}a^{4}+\frac{191496491}{4143048}a^{3}+\frac{4110425}{147966}a^{2}-\frac{4882393}{591864}a-\frac{176071}{42276}$, $\frac{31803}{690508}a^{15}+\frac{2523}{98644}a^{14}-\frac{2249825}{4143048}a^{13}-\frac{60349}{197288}a^{12}+\frac{325447}{517881}a^{11}+\frac{37099}{98644}a^{10}+\frac{15154117}{1381016}a^{9}+\frac{1214817}{197288}a^{8}-\frac{56365105}{2071524}a^{7}-\frac{1536979}{98644}a^{6}-\frac{64556071}{1381016}a^{5}-\frac{5120985}{197288}a^{4}+\frac{130899179}{1035762}a^{3}+\frac{6999221}{98644}a^{2}-\frac{8051635}{591864}a-\frac{240341}{28184}$ 99/14092a^14 - 2155/42276a^12 - 5519/42276a^10 + 14415/14092a^8 + 23573/42276a^6 - 48265/14092a^4 + 38269/42276a^2 - 20015/42276, 2677/591864a^14 - 25355/591864a^12 - 5951/591864a^10 + 510637/591864a^8 - 777619/591864a^6 - 1984795/591864a^4 + 1776443/197288a^2 - 60653/84552, 20/517881a^15 - 13801/1035762a^13 + 31823/345254a^11 + 536789/2071524a^9 - 326153/172627a^7 - 1183123/1035762a^5 + 7231349/1035762a^3 - 261819/98644a, 103009/2071524a^15 - 1244011/2071524a^13 + 553439/690508a^11 + 6196594/517881a^9 - 22002693/690508a^7 - 101269111/2071524a^5 + 303592109/2071524a^3 - 572366/24661a, 32503/4143048a^15 - 401/73983a^14 - 53423/1035762a^13 + 41485/591864a^12 - 801251/4143048a^11 - 10993/98644a^10 + 568613/517881a^9 - 861787/591864a^8 + 6752051/4143048a^7 + 198295/49322a^6 - 2307161/517881a^5 + 3946561/591864a^4 - 4847341/1381016a^3 - 5490265/295932a^2 + 159139/147966a + 69805/28184, 32503/4143048a^15 + 401/73983a^14 - 53423/1035762a^13 - 41485/591864a^12 - 801251/4143048a^11 + 10993/98644a^10 + 568613/517881a^9 + 861787/591864a^8 + 6752051/4143048a^7 - 198295/49322a^6 - 2307161/517881a^5 - 3946561/591864a^4 - 4847341/1381016a^3 + 5490265/295932a^2 + 159139/147966a - 69805/28184, 38023/1381016a^15 + 7127/591864a^14 - 1348507/4143048a^13 - 74891/591864a^12 + 1559251/4143048a^11 + 9747/197288a^10 + 9133469/1381016a^9 + 1525403/591864a^8 - 67636531/4143048a^7 - 920883/197288a^6 - 39656241/1381016a^5 - 6485939/591864a^4 + 314828635/4143048a^3 + 13383745/591864a^2 - 3145463/591864a - 22313/28184, 4603/147966a^14 - 7498/24661a^12 - 2348/73983a^10 + 914059/147966a^8 - 1298819/147966a^6 - 1874876/73983a^4 + 3486890/73983a^2 - 106891/21138, 1397/98644a^14 - 33541/295932a^12 - 58307/295932a^10 + 116385/49322a^8 - 116737/295932a^6 - 919071/98644a^4 + 2372761/295932a^2 - 19363/21138, 150145/4143048a^15 + 2001/197288a^14 - 132575/345254a^13 - 5071/295932a^12 + 779573/4143048a^11 - 362431/591864a^10 + 15995537/2071524a^9 + 47195/98644a^8 - 62014787/4143048a^7 + 5671711/591864a^6 - 16148450/517881a^5 - 115879/98644a^4 + 307459489/4143048a^3 - 19436779/591864a^2 - 3874631/295932a + 231661/42276, 275/39837a^15 - 24727/591864a^14 - 13817/79674a^13 + 179369/591864a^12 + 61153/79674a^11 + 479963/591864a^10 + 134711/39837a^9 - 3761605/591864a^8 - 722570/39837a^7 - 2287751/591864a^6 - 1159759/79674a^5 + 15245665/591864a^4 + 1998895/26558a^3 - 723223/197288a^2 - 76576/5691a - 118867/84552, 12829/1381016a^15 - 4961/295932a^14 - 586577/4143048a^13 + 3629/49322a^12 + 1535123/4143048a^11 + 99787/147966a^10 + 3817587/1381016a^9 - 473393/295932a^8 - 43910777/4143048a^7 - 2644301/295932a^6 - 15468779/1381016a^5 + 808663/147966a^4 + 191496491/4143048a^3 + 4110425/147966a^2 - 4882393/591864a - 176071/42276, 31803/690508a^15 + 2523/98644a^14 - 2249825/4143048a^13 - 60349/197288a^12 + 325447/517881a^11 + 37099/98644a^10 + 15154117/1381016a^9 + 1214817/197288a^8 - 56365105/2071524a^7 - 1536979/98644a^6 - 64556071/1381016a^5 - 5120985/197288a^4 + 130899179/1035762a^3 + 6999221/98644a^2 - 8051635/591864*a - 240341/28184 (assuming GRH)	comment:Fundamental units 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:	$ 61882651.1808 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 61882651.1808 \cdot 1}{2\cdot\sqrt{121821863230510940507078656}}\cr\approx \mathstrut & 0.453310711887 \end{aligned}\] (assuming GRH)

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 16*x^12 + 236*x^10 - 638*x^8 - 908*x^6 + 2944*x^4 - 756*x^2 + 49) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - 12*x^14 + 16*x^12 + 236*x^10 - 638*x^8 - 908*x^6 + 2944*x^4 - 756*x^2 + 49, 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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 16*x^12 + 236*x^10 - 638*x^8 - 908*x^6 + 2944*x^4 - 756*x^2 + 49); 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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 12*x^14 + 16*x^12 + 236*x^10 - 638*x^8 - 908*x^6 + 2944*x^4 - 756*x^2 + 49); 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

$C_2\wr D_4$ (as 16T1439):

comment:Galois group

sage:K.galois_group(type='pari')

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A solvable group of order 2048

The 74 conjugacy class representatives for $C_2\wr D_4$

Character table for $C_2\wr D_4$

Intermediate fields

$\Q(\sqrt{7}) $, $\Q(\sqrt{2}) $, $\Q(\sqrt{14}) $, $\Q(\sqrt{2}, \sqrt{7})$, 8.8.86228860928.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(L)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Degree 16 siblings:	data not computed
Degree 32 siblings:	data not computed
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	${\href{/padicField/3.4.0.1}{4} }^{4}$	${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$	R	${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$	${\href{/padicField/17.4.0.1}{4} }^{4}$	${\href{/padicField/19.4.0.1}{4} }^{4}$	${\href{/padicField/23.4.0.1}{4} }^{4}$	${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$	${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.4.0.1}{4} }^{4}$	${\href{/padicField/41.4.0.1}{4} }^{4}$	${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$	${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$	${\href{/padicField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
$2$ 2	2.1.16.50h1.45	$x^{16} + 2 x^{12} + 8 x^{11} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 8 x^{3} + 2$	$16$	$1$	$50$	16T345	$$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{2}$$
$7$ 7	7.1.2.1a1.2	$x^{2} + 21$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	7.1.2.1a1.2	$x^{2} + 21$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	7.2.2.2a1.2	$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
	7.4.2.4a1.2	$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$	$2$	$4$	$4$	$C_4\times C_2$	$$[\ ]_{2}^{4}$$
$137$ 137	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	137.2.1.0a1.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	137.1.2.1a1.1	$x^{2} + 137$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	137.2.1.0a1.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	137.2.1.0a1.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	137.1.2.1a1.1	$x^{2} + 137$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	137.2.1.0a1.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$

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