Properties

Label 1800s3
Conductor 18001800
Discriminant 3499200000034992000000
j-invariant 287562283 \frac{28756228}{3}
CM no
Rank 11
Torsion structure Z/2Z\Z/{2}\Z

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2=x314475x670250y^2=x^3-14475x-670250 Copy content Toggle raw display (homogenize, simplify)
y2z=x314475xz2670250z3y^2z=x^3-14475xz^2-670250z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x314475x670250y^2=x^3-14475x-670250 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([0, 0, 0, -14475, -670250])
 
Copy content gp:E = ellinit([0, 0, 0, -14475, -670250])
 
Copy content magma:E := EllipticCurve([0, 0, 0, -14475, -670250]);
 
Copy content oscar:E = elliptic_curve([0, 0, 0, -14475, -670250])
 
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

ZZ/2Z\Z \oplus \Z/{2}\Z

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

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(69,4)(-69, 4)1.66593034773173178998483328731.6659303477317317899848332873\infty
(70,0)(-70, 0)0022

Integral points

(70,0) \left(-70, 0\right) , (69,±4)(-69,\pm 4), (155,±900)(155,\pm 900), (714,±18788)(714,\pm 18788) Copy content Toggle raw display

Copy content comment:Integral points
 
Copy content sage:E.integral_points()
 
Copy content magma:IntegralPoints(E);
 

Invariants

Conductor: NN  =  1800 1800  = 2332522^{3} \cdot 3^{2} \cdot 5^{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  =  3499200000034992000000 = 21037562^{10} \cdot 3^{7} \cdot 5^{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  =  287562283 \frac{28756228}{3}  = 223119332^{2} \cdot 3^{-1} \cdot 193^{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}} ≈ 1.05524640475122041504760766901.0552464047512204150476076690
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}} ≈ 0.87640134626650570913142138396-0.87640134626650570913142138396
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.05617163459916921.0561716345991692
Szpiro ratio: σm\sigma_{m} ≈ 5.383751867731585.38375186773158

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
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 = 1 1
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) ≈ 1.66593034773173178998483328731.6659303477317317899848332873
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 ≈ 0.435258870003495234190049595390.43525887000349523419004959539
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  = 2222 2\cdot2^{2}\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}} = 22
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) ≈ 2.90044384263297383504603605662.9004438426329738350460360566
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    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

2.900443843L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.4352591.66593016222.900443843\begin{aligned} 2.900443843 \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 0.435259 \cdot 1.665930 \cdot 16}{2^2} \\ & \approx 2.900443843\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, 0, 0, -14475, -670250]); 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, 0, 0, -14475, -670250]); 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   1800.2.a.m

q4q11+2q13+2q174q19+O(q20) q - 4 q^{11} + 2 q^{13} + 2 q^{17} - 4 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: 2048
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: no
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 3 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 22 IIIIII^{*} additive -1 3 10 0
33 44 I1I_{1}^{*} additive -1 2 7 1
55 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 except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
22 2B 16.48.0.127

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 = [[5, 4, 236, 237], [143, 0, 0, 239], [225, 16, 224, 17], [1, 0, 16, 1], [15, 2, 142, 227], [61, 160, 60, 121], [1, 16, 0, 1], [106, 115, 95, 126], [64, 235, 45, 14]] GL(2,Integers(240)).subgroup(gens)
 
Copy content magma:Gens := [[5, 4, 236, 237], [143, 0, 0, 239], [225, 16, 224, 17], [1, 0, 16, 1], [15, 2, 142, 227], [61, 160, 60, 121], [1, 16, 0, 1], [106, 115, 95, 126], [64, 235, 45, 14]]; sub<GL(2,Integers(240))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 240=2435 240 = 2^{4} \cdot 3 \cdot 5 , index 192192, genus 11, and generators

(54236237),(14300239),(2251622417),(10161),(152142227),(6116060121),(11601),(10611595126),(642354514)\left(\begin{array}{rr} 5 & 4 \\ 236 & 237 \end{array}\right),\left(\begin{array}{rr} 143 & 0 \\ 0 & 239 \end{array}\right),\left(\begin{array}{rr} 225 & 16 \\ 224 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 142 & 227 \end{array}\right),\left(\begin{array}{rr} 61 & 160 \\ 60 & 121 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 106 & 115 \\ 95 & 126 \end{array}\right),\left(\begin{array}{rr} 64 & 235 \\ 45 & 14 \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[240])K:=\Q(E[240]) is a degree-29491202949120 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/240Z)\GL_2(\Z/240\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 22 225=3252 225 = 3^{2} \cdot 5^{2}
33 additive 88 200=2352 200 = 2^{3} \cdot 5^{2}
55 additive 1414 72=2332 72 = 2^{3} \cdot 3^{2}

Isogenies

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

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2, 4 and 8.
Its isogeny class 1800s consists of 6 curves linked by isogenies of degrees dividing 8.

Twists

The minimal quadratic twist of this elliptic curve is 24a3, its twist by 15-15.

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} Z/2Z\cong \Z/{2}\Z are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(3)\Q(\sqrt{3}) Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
22 Q(5)\Q(\sqrt{-5}) Z/4Z\Z/4\Z not in database
22 Q(15)\Q(\sqrt{-15}) Z/4Z\Z/4\Z not in database
44 Q(3,5)\Q(\sqrt{3}, \sqrt{-5}) Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
44 Q(2,15)\Q(\sqrt{2}, \sqrt{-15}) Z/8Z\Z/8\Z not in database
44 Q(6,10)\Q(\sqrt{6}, \sqrt{-10}) Z/8Z\Z/8\Z not in database
88 8.4.1866240000.3 Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
88 8.0.29859840000.8 Z/8Z\Z/8\Z not in database
88 8.0.3317760000.6 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
88 8.2.113374080000.3 Z/6Z\Z/6\Z not in database
1616 16.0.3482851737600000000.2 Z/4ZZ/4Z\Z/4\Z \oplus \Z/4\Z not in database
1616 deg 16 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
1616 16.0.891610044825600000000.11 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
1616 deg 16 Z/16Z\Z/16\Z not in database
1616 16.0.58432555897690521600000000.3 Z/2ZZ/16Z\Z/2\Z \oplus \Z/16\Z not in database
1616 deg 16 Z/2ZZ/6Z\Z/2\Z \oplus \Z/6\Z not in database
1616 deg 16 Z/12Z\Z/12\Z not in database
1616 deg 16 Z/12Z\Z/12\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

pp 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add add add ss ord ord ord ord ord ord ord ord ord ord ss
λ\lambda-invariant(s) - - - 1,1 1 3 5 1 1 1 1 1 1 1 1,1
μ\mu-invariant(s) - - - 0,0 0 0 0 0 0 0 0 0 0 0 0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

pp-adic regulators

pp-adic regulators are not yet computed for curves that are not Γ0\Gamma_0-optimal.