Properties

Label 383496t1
Conductor 383496383496
Discriminant 9.400×1015-9.400\times 10^{15}
j-invariant 7057510461731 \frac{70575104}{61731}
CM no
Rank 00
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2=x3+x2+45975x+2728827y^2=x^3+x^2+45975x+2728827 Copy content Toggle raw display (homogenize, simplify)
y2z=x3+x2z+45975xz2+2728827z3y^2z=x^3+x^2z+45975xz^2+2728827z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3+3723948x+1978143012y^2=x^3+3723948x+1978143012 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([0, 1, 0, 45975, 2728827])
 
Copy content gp:E = ellinit([0, 1, 0, 45975, 2728827])
 
Copy content magma:E := EllipticCurve([0, 1, 0, 45975, 2728827]);
 
Copy content oscar:E = elliptic_curve([0, 1, 0, 45975, 2728827])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

trivial

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Invariants

Conductor: NN  =  383496 383496  = 233192922^{3} \cdot 3 \cdot 19 \cdot 29^{2}
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Discriminant: Δ\Delta  =  9400073837734656-9400073837734656 = 12832193296-1 \cdot 2^{8} \cdot 3^{2} \cdot 19^{3} \cdot 29^{6}
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: jj  =  7057510461731 \frac{70575104}{61731}  = 210321934132^{10} \cdot 3^{-2} \cdot 19^{-3} \cdot 41^{3}
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 1.75245264377787052397325404541.7524526437778705239732540454
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.39329339158866336256320338509-0.39329339158866336256320338509
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.95058130979447650.9505813097944765
Szpiro ratio: σm\sigma_{m} ≈ 3.40832640044544373.4083264004454437

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 0 0
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: rr = 0 0
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) = 11
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: Ω\Omega ≈ 0.266539137193805401417285365790.26653913719380540141728536579
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 16 16  = 22212 2^{2}\cdot2\cdot1\cdot2
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 11
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: L(E,1) L(E,1) ≈ 4.26462619510088642267656585264.2646261951008864226765658526
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  =  11    (exact)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

4.264626195L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.2665391.00000016124.264626195\begin{aligned} 4.264626195 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.266539 \cdot 1.000000 \cdot 16}{1^2} \\ & \approx 4.264626195\end{aligned}

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([0, 1, 0, 45975, 2728827]); r = E.rank(); ar = E.analytic_rank(); assert r == ar; Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical(); omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order(); assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([0, 1, 0, 45975, 2728827]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar; sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1); reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E); assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form 383496.2.a.t

q+q3+q53q7+q9+5q112q13+q15+q17q19+O(q20) q + q^{3} + q^{5} - 3 q^{7} + q^{9} + 5 q^{11} - 2 q^{13} + q^{15} + q^{17} - q^{19} + O(q^{20}) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:\\ actual modular form, use for small N [mf,F] = mffromell(E) Ser(mfcoefs(mf,20),q) \\ or just the series Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 2322432
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
22 44 I1I_{1}^{*} additive 1 3 8 0
33 22 I2I_{2} split multiplicative -1 1 2 2
1919 11 I3I_{3} nonsplit multiplicative 1 1 3 3
2929 22 I0I_0^{*} additive 1 2 6 0

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content oscar:[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell.

Copy content comment:Mod p Galois image
 
Copy content sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
Copy content magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[37, 2, 36, 3], [1, 2, 0, 1], [1, 0, 2, 1], [21, 2, 21, 3], [1, 1, 37, 0]] GL(2,Integers(38)).subgroup(gens)
 
Copy content magma:Gens := [[37, 2, 36, 3], [1, 2, 0, 1], [1, 0, 2, 1], [21, 2, 21, 3], [1, 1, 37, 0]]; sub<GL(2,Integers(38))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has label 38.2.0.a.1, level 38=219 38 = 2 \cdot 19 , index 22, genus 00, and generators

(372363),(1201),(1021),(212213),(11370)\left(\begin{array}{rr} 37 & 2 \\ 36 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 2 \\ 21 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 37 & 0 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[38])K:=\Q(E[38]) is a degree-369360369360 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/38Z)\GL_2(\Z/38\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
22 additive 22 15979=19292 15979 = 19 \cdot 29^{2}
33 split multiplicative 44 6728=23292 6728 = 2^{3} \cdot 29^{2}
1919 nonsplit multiplicative 2020 20184=233292 20184 = 2^{3} \cdot 3 \cdot 29^{2}
2929 additive 422422 456=23319 456 = 2^{3} \cdot 3 \cdot 19

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 383496t consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 456d1, its twist by 2929.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} (which is trivial) are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
33 3.1.76.1 Z/2Z\Z/2\Z not in database
66 6.0.109744.2 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
88 deg 8 Z/3Z\Z/3\Z not in database
1212 deg 12 Z/4Z\Z/4\Z not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

pp-adic regulators

All pp-adic regulators are identically 11 since the rank is 00.