LMFDB - Number field 16.0.119842600295694598144.8

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 + 8*x^13 - 40*x^12 + 56*x^11 - 24*x^10 - 112*x^9 + 383*x^8 - 598*x^7 + 723*x^6 - 544*x^5 + 384*x^4 - 104*x^3 + 92*x^2 + 32)

gp: K = bnfinit(y^16 - 2*y^15 + y^14 + 8*y^13 - 40*y^12 + 56*y^11 - 24*y^10 - 112*y^9 + 383*y^8 - 598*y^7 + 723*y^6 - 544*y^5 + 384*y^4 - 104*y^3 + 92*y^2 + 32, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + x^14 + 8*x^13 - 40*x^12 + 56*x^11 - 24*x^10 - 112*x^9 + 383*x^8 - 598*x^7 + 723*x^6 - 544*x^5 + 384*x^4 - 104*x^3 + 92*x^2 + 32);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + x^14 + 8*x^13 - 40*x^12 + 56*x^11 - 24*x^10 - 112*x^9 + 383*x^8 - 598*x^7 + 723*x^6 - 544*x^5 + 384*x^4 - 104*x^3 + 92*x^2 + 32)

$ x^{16} - 2 x^{15} + x^{14} + 8 x^{13} - 40 x^{12} + 56 x^{11} - 24 x^{10} - 112 x^{9} + 383 x^{8} + \cdots + 32 $ x^16 - 2*x^15 + x^14 + 8*x^13 - 40*x^12 + 56*x^11 - 24*x^10 - 112*x^9 + 383*x^8 - 598*x^7 + 723*x^6 - 544*x^5 + 384*x^4 - 104*x^3 + 92*x^2 + 32

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$16$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 8]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$119842600295694598144$ 119842600295694598144 $\medspace = 2^{34}\cdot 17^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$17.99$	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:	$2^{11/4}17^{1/2}\approx 27.736837922354518$
Ramified primes:	$2$, $17$ 2, 17	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{11}+\frac{1}{16}a^{10}+\frac{1}{16}a^{9}+\frac{1}{16}a^{8}+\frac{3}{16}a^{7}-\frac{1}{16}a^{6}-\frac{3}{16}a^{5}-\frac{3}{16}a^{4}+\frac{1}{8}a^{3}-\frac{1}{8}a^{2}$, $\frac{1}{48}a^{12}-\frac{1}{12}a^{10}+\frac{1}{12}a^{9}-\frac{1}{24}a^{8}-\frac{1}{6}a^{7}-\frac{1}{8}a^{6}-\frac{1}{12}a^{5}+\frac{17}{48}a^{4}-\frac{1}{2}a^{3}-\frac{7}{24}a^{2}+\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{96}a^{13}-\frac{1}{96}a^{12}+\frac{1}{48}a^{11}+\frac{1}{48}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{12}a^{7}+\frac{1}{12}a^{6}-\frac{11}{32}a^{5}+\frac{25}{96}a^{4}+\frac{5}{48}a^{3}+\frac{17}{48}a^{2}-\frac{1}{6}a-\frac{1}{6}$, $\frac{1}{96}a^{14}-\frac{1}{96}a^{12}-\frac{1}{48}a^{11}-\frac{1}{12}a^{10}+\frac{5}{48}a^{9}-\frac{1}{16}a^{8}+\frac{7}{48}a^{7}-\frac{7}{96}a^{6}-\frac{5}{16}a^{5}+\frac{19}{96}a^{4}-\frac{1}{6}a^{3}-\frac{19}{48}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{299263584}a^{15}+\frac{276121}{149631792}a^{14}+\frac{458825}{149631792}a^{13}-\frac{1882277}{299263584}a^{12}+\frac{106565}{9351987}a^{11}+\frac{4000889}{49877264}a^{10}+\frac{244015}{3117329}a^{9}+\frac{725697}{12469316}a^{8}+\frac{26764151}{299263584}a^{7}+\frac{92377}{149631792}a^{6}+\frac{8684187}{49877264}a^{5}-\frac{97895479}{299263584}a^{4}-\frac{24940433}{74815896}a^{3}-\frac{28382999}{149631792}a^{2}-\frac{1412115}{3117329}a-\frac{2963191}{18703974}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^2, 1/2*a^7 - 1/2*a^3, 1/2*a^8 - 1/2*a^4, 1/4*a^9 - 1/4*a^7 - 1/4*a^5 - 1/4*a^3 - 1/2*a, 1/4*a^10 - 1/4*a^8 - 1/4*a^6 - 1/4*a^4 - 1/2*a^2, 1/16*a^11 + 1/16*a^10 + 1/16*a^9 + 1/16*a^8 + 3/16*a^7 - 1/16*a^6 - 3/16*a^5 - 3/16*a^4 + 1/8*a^3 - 1/8*a^2, 1/48*a^12 - 1/12*a^10 + 1/12*a^9 - 1/24*a^8 - 1/6*a^7 - 1/8*a^6 - 1/12*a^5 + 17/48*a^4 - 1/2*a^3 - 7/24*a^2 + 1/6*a + 1/3, 1/96*a^13 - 1/96*a^12 + 1/48*a^11 + 1/48*a^10 - 1/8*a^9 - 1/8*a^8 + 1/12*a^7 + 1/12*a^6 - 11/32*a^5 + 25/96*a^4 + 5/48*a^3 + 17/48*a^2 - 1/6*a - 1/6, 1/96*a^14 - 1/96*a^12 - 1/48*a^11 - 1/12*a^10 + 5/48*a^9 - 1/16*a^8 + 7/48*a^7 - 7/96*a^6 - 5/16*a^5 + 19/96*a^4 - 1/6*a^3 - 19/48*a^2 - 1/2*a - 1/2, 1/299263584*a^15 + 276121/149631792*a^14 + 458825/149631792*a^13 - 1882277/299263584*a^12 + 106565/9351987*a^11 + 4000889/49877264*a^10 + 244015/3117329*a^9 + 725697/12469316*a^8 + 26764151/299263584*a^7 + 92377/149631792*a^6 + 8684187/49877264*a^5 - 97895479/299263584*a^4 - 24940433/74815896*a^3 - 28382999/149631792*a^2 - 1412115/3117329*a - 2963191/18703974

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

Trivial group, which has order $1$

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:	$7$	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:	$\frac{725027}{9653664}a^{15}-\frac{630067}{3217888}a^{14}+\frac{160301}{1608944}a^{13}+\frac{1691515}{2413416}a^{12}-\frac{1366593}{402236}a^{11}+\frac{13044587}{2413416}a^{10}-\frac{3904403}{2413416}a^{9}-\frac{26503957}{2413416}a^{8}+\frac{109088043}{3217888}a^{7}-\frac{510929051}{9653664}a^{6}+\frac{268747721}{4826832}a^{5}-\frac{89909335}{2413416}a^{4}+\frac{21813797}{1206708}a^{3}-\frac{3770209}{804472}a^{2}+\frac{191985}{201118}a-\frac{413108}{301677}$, $\frac{4684729}{299263584}a^{15}-\frac{1515301}{299263584}a^{14}-\frac{1373825}{37407948}a^{13}+\frac{708577}{6234658}a^{12}-\frac{17738545}{49877264}a^{11}-\frac{22325975}{149631792}a^{10}+\frac{36199879}{49877264}a^{9}-\frac{52044105}{49877264}a^{8}+\frac{548774629}{299263584}a^{7}-\frac{94377725}{299263584}a^{6}+\frac{42598265}{49877264}a^{5}-\frac{43643087}{49877264}a^{4}+\frac{43039858}{9351987}a^{3}-\frac{6457972}{3117329}a^{2}+\frac{19483450}{9351987}a+\frac{4569179}{9351987}$, $\frac{3056919}{99754528}a^{15}+\frac{7308265}{149631792}a^{14}-\frac{18478867}{99754528}a^{13}+\frac{12799799}{49877264}a^{12}-\frac{6804509}{37407948}a^{11}-\frac{130667875}{49877264}a^{10}+\frac{656816105}{149631792}a^{9}-\frac{371382383}{149631792}a^{8}-\frac{409342185}{99754528}a^{7}+\frac{1604710945}{74815896}a^{6}-\frac{8729891789}{299263584}a^{5}+\frac{193258427}{6234658}a^{4}-\frac{1588569091}{149631792}a^{3}+\frac{315459677}{37407948}a^{2}+\frac{55857763}{18703974}a+\frac{38944327}{9351987}$, $\frac{20023439}{149631792}a^{15}-\frac{1022316}{3117329}a^{14}+\frac{27121663}{149631792}a^{13}+\frac{170666893}{149631792}a^{12}-\frac{872624953}{149631792}a^{11}+\frac{461370007}{49877264}a^{10}-\frac{575006251}{149631792}a^{9}-\frac{827949651}{49877264}a^{8}+\frac{179013799}{3117329}a^{7}-\frac{4672439901}{49877264}a^{6}+\frac{2695743067}{24938632}a^{5}-\frac{6302323283}{74815896}a^{4}+\frac{979924087}{18703974}a^{3}-\frac{360562793}{18703974}a^{2}+\frac{140966891}{18703974}a-\frac{35373067}{9351987}$, $\frac{4610777}{149631792}a^{15}-\frac{98557}{18703974}a^{14}-\frac{11908399}{149631792}a^{13}+\frac{9296473}{37407948}a^{12}-\frac{106103317}{149631792}a^{11}-\frac{71118275}{149631792}a^{10}+\frac{280361525}{149631792}a^{9}-\frac{448886587}{149631792}a^{8}+\frac{101045411}{24938632}a^{7}+\frac{258205651}{149631792}a^{6}-\frac{41928560}{9351987}a^{5}+\frac{1287505901}{149631792}a^{4}-\frac{14668543}{9351987}a^{3}+\frac{60104597}{24938632}a^{2}+\frac{72004205}{18703974}a+\frac{2426602}{3117329}$, $\frac{6333787}{149631792}a^{15}-\frac{6452009}{149631792}a^{14}-\frac{7285357}{149631792}a^{13}+\frac{13900621}{37407948}a^{12}-\frac{201728167}{149631792}a^{11}+\frac{102064477}{149631792}a^{10}+\frac{213522755}{149631792}a^{9}-\frac{272840325}{49877264}a^{8}+\frac{862403719}{74815896}a^{7}-\frac{351435997}{37407948}a^{6}+\frac{66090131}{12469316}a^{5}+\frac{924376541}{149631792}a^{4}-\frac{25008930}{3117329}a^{3}+\frac{983230333}{74815896}a^{2}-\frac{90010253}{18703974}a+\frac{42041650}{9351987}$, $\frac{4254677}{18703974}a^{15}-\frac{7661781}{12469316}a^{14}+\frac{940500}{3117329}a^{13}+\frac{20206664}{9351987}a^{12}-\frac{32570090}{3117329}a^{11}+\frac{157525822}{9351987}a^{10}-\frac{42858196}{9351987}a^{9}-\frac{317634266}{9351987}a^{8}+\frac{650537017}{6234658}a^{7}-\frac{6094785217}{37407948}a^{6}+\frac{1576140560}{9351987}a^{5}-\frac{1081135223}{9351987}a^{4}+\frac{498215450}{9351987}a^{3}-\frac{49689355}{3117329}a^{2}+\frac{6940064}{3117329}a-\frac{33535553}{9351987}$ 725027/9653664a^15 - 630067/3217888a^14 + 160301/1608944a^13 + 1691515/2413416a^12 - 1366593/402236a^11 + 13044587/2413416a^10 - 3904403/2413416a^9 - 26503957/2413416a^8 + 109088043/3217888a^7 - 510929051/9653664a^6 + 268747721/4826832a^5 - 89909335/2413416a^4 + 21813797/1206708a^3 - 3770209/804472a^2 + 191985/201118a - 413108/301677, 4684729/299263584a^15 - 1515301/299263584a^14 - 1373825/37407948a^13 + 708577/6234658a^12 - 17738545/49877264a^11 - 22325975/149631792a^10 + 36199879/49877264a^9 - 52044105/49877264a^8 + 548774629/299263584a^7 - 94377725/299263584a^6 + 42598265/49877264a^5 - 43643087/49877264a^4 + 43039858/9351987a^3 - 6457972/3117329a^2 + 19483450/9351987a + 4569179/9351987, 3056919/99754528a^15 + 7308265/149631792a^14 - 18478867/99754528a^13 + 12799799/49877264a^12 - 6804509/37407948a^11 - 130667875/49877264a^10 + 656816105/149631792a^9 - 371382383/149631792a^8 - 409342185/99754528a^7 + 1604710945/74815896a^6 - 8729891789/299263584a^5 + 193258427/6234658a^4 - 1588569091/149631792a^3 + 315459677/37407948a^2 + 55857763/18703974a + 38944327/9351987, 20023439/149631792a^15 - 1022316/3117329a^14 + 27121663/149631792a^13 + 170666893/149631792a^12 - 872624953/149631792a^11 + 461370007/49877264a^10 - 575006251/149631792a^9 - 827949651/49877264a^8 + 179013799/3117329a^7 - 4672439901/49877264a^6 + 2695743067/24938632a^5 - 6302323283/74815896a^4 + 979924087/18703974a^3 - 360562793/18703974a^2 + 140966891/18703974a - 35373067/9351987, 4610777/149631792a^15 - 98557/18703974a^14 - 11908399/149631792a^13 + 9296473/37407948a^12 - 106103317/149631792a^11 - 71118275/149631792a^10 + 280361525/149631792a^9 - 448886587/149631792a^8 + 101045411/24938632a^7 + 258205651/149631792a^6 - 41928560/9351987a^5 + 1287505901/149631792a^4 - 14668543/9351987a^3 + 60104597/24938632a^2 + 72004205/18703974a + 2426602/3117329, 6333787/149631792a^15 - 6452009/149631792a^14 - 7285357/149631792a^13 + 13900621/37407948a^12 - 201728167/149631792a^11 + 102064477/149631792a^10 + 213522755/149631792a^9 - 272840325/49877264a^8 + 862403719/74815896a^7 - 351435997/37407948a^6 + 66090131/12469316a^5 + 924376541/149631792a^4 - 25008930/3117329a^3 + 983230333/74815896a^2 - 90010253/18703974a + 42041650/9351987, 4254677/18703974a^15 - 7661781/12469316a^14 + 940500/3117329a^13 + 20206664/9351987a^12 - 32570090/3117329a^11 + 157525822/9351987a^10 - 42858196/9351987a^9 - 317634266/9351987a^8 + 650537017/6234658a^7 - 6094785217/37407948a^6 + 1576140560/9351987a^5 - 1081135223/9351987a^4 + 498215450/9351987a^3 - 49689355/3117329a^2 + 6940064/3117329*a - 33535553/9351987	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:	$ 76945.8851596 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 76945.8851596 \cdot 1}{2\cdot\sqrt{119842600295694598144}}\cr\approx \mathstrut & 8.53667483673 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 + 8*x^13 - 40*x^12 + 56*x^11 - 24*x^10 - 112*x^9 + 383*x^8 - 598*x^7 + 723*x^6 - 544*x^5 + 384*x^4 - 104*x^3 + 92*x^2 + 32)

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^16 - 2*x^15 + x^14 + 8*x^13 - 40*x^12 + 56*x^11 - 24*x^10 - 112*x^9 + 383*x^8 - 598*x^7 + 723*x^6 - 544*x^5 + 384*x^4 - 104*x^3 + 92*x^2 + 32, 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^16 - 2*x^15 + x^14 + 8*x^13 - 40*x^12 + 56*x^11 - 24*x^10 - 112*x^9 + 383*x^8 - 598*x^7 + 723*x^6 - 544*x^5 + 384*x^4 - 104*x^3 + 92*x^2 + 32);

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^16 - 2*x^15 + x^14 + 8*x^13 - 40*x^12 + 56*x^11 - 24*x^10 - 112*x^9 + 383*x^8 - 598*x^7 + 723*x^6 - 544*x^5 + 384*x^4 - 104*x^3 + 92*x^2 + 32);

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

$C_2^3:C_4$ (as 16T33):

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 32

The 11 conjugacy class representatives for $C_2^3:C_4$

Character table for $C_2^3:C_4$

Intermediate fields

$\Q(\sqrt{2}) $, $\Q(\sqrt{34}) $, $\Q(\sqrt{17}) $, 4.0.2312.1 x2, $\Q(\sqrt{2}, \sqrt{17})$, 4.0.1088.2 x2, 8.0.1368408064.3 x2, 8.0.342102016.2, 8.4.10947264512.1 x2

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

Galois closure:	deg 32
Degree 8 siblings:	8.4.1212153856.4, 8.4.10947264512.1, 8.0.1368408064.3, 8.0.350312464384.4
Degree 16 siblings:	16.8.122718822702791268499456.3, deg 16, 16.0.424632604507928264704.1
Minimal sibling:	8.0.1368408064.3

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.4.0.1}{4} }^{4}$	${\href{/padicField/5.4.0.1}{4} }^{4}$	${\href{/padicField/7.4.0.1}{4} }^{4}$	${\href{/padicField/11.4.0.1}{4} }^{4}$	${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$	R	${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$	${\href{/padicField/23.4.0.1}{4} }^{4}$	${\href{/padicField/29.4.0.1}{4} }^{4}$	${\href{/padicField/31.2.0.1}{2} }^{8}$	${\href{/padicField/37.4.0.1}{4} }^{4}$	${\href{/padicField/41.4.0.1}{4} }^{4}$	${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$	${\href{/padicField/47.2.0.1}{2} }^{8}$	${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$	${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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
$2$ 2	2.2.3.1	$x^{2} + 4 x + 2$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.1	$x^{2} + 4 x + 2$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.1	$x^{2} + 4 x + 2$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.1	$x^{2} + 4 x + 2$	$2$	$1$	$3$	$C_2$	$[3]$
	2.4.11.1	$x^{4} + 8 x^{3} + 4 x^{2} + 2$	$4$	$1$	$11$	$C_4$	$[3, 4]$
	2.4.11.1	$x^{4} + 8 x^{3} + 4 x^{2} + 2$	$4$	$1$	$11$	$C_4$	$[3, 4]$
$17$ 17	17.4.2.1	$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	17.4.2.1	$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	17.4.2.1	$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	17.4.2.1	$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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