LMFDB - Elliptic curve with Cremona label 438702f1 (LMFDB label 438702.f1)

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

$y^2+xy=x^3+x^2-5161401x-4729230171$ y^2+xy=x^3+x^2-5161401x-4729230171	(homogenize, simplify)
$y^2z+xyz=x^3+x^2z-5161401xz^2-4729230171z^3$ y^2z+xyz=x^3+x^2z-5161401xz^2-4729230171z^3	(dehomogenize, simplify)
$y^2=x^3-6689176371x-220546625215986$ y^2=x^3-6689176371x-220546625215986	(homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 1, 0, -5161401, -4729230171])

gp: E = ellinit([1, 1, 0, -5161401, -4729230171])

magma: E := EllipticCurve([1, 1, 0, -5161401, -4729230171]);

oscar: E = elliptic_curve([1, 1, 0, -5161401, -4729230171])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(29558479/10201, 68796240926/1030301)$	$13.146769961001791169505742787$	$\infty$

Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

Invariants

Conductor:	$N$	=	$438702$	=	$2 \cdot 3 \cdot 11 \cdot 17^{2} \cdot 23$	comment: Conductor sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); oscar: conductor(E)
Discriminant:	$\Delta$	=	$-853156748497930956288$	=	$-1 \cdot 2^{9} \cdot 3^{3} \cdot 11^{3} \cdot 17^{10} \cdot 23$	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	$j$	=	$-\frac{629989223007953593}{35345595428352}$	=	$-1 \cdot 2^{-9} \cdot 3^{-3} \cdot 11^{-3} \cdot 17^{-4} \cdot 23^{-1} \cdot 773^{3} \cdot 1109^{3}$	comment: j-invariant sage: E.j_invariant().factor() gp: E.j magma: jInvariant(E); oscar: j_invariant(E)
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$ (no potential complex multiplication)			sage: E.has_cm() magma: HasComplexMultiplication(E);
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$2.7736453696613341370137226893$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$1.3570386976332260968889553804$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$Q$	≈	$0.9260263665058911$
Szpiro ratio:	$\sigma_{m}$	≈	$4.470303985477481$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$	sage: E.analytic_rank() gp: ellanalyticrank(E) magma: AnalyticRank(E);
Mordell-Weil rank:	$r$	=	$1$	comment: Rank sage: E.rank() gp: [lower,upper] = ellrank(E) magma: Rank(E);
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$13.146769961001791169505742787$	comment: Regulator sage: E.regulator() G = E.gen \\ if available matdet(ellheightmatrix(E,G)) magma: Regulator(E);
Real period:	$\Omega$	≈	$0.049918898467759642360589662946$	comment: Real Period sage: E.period_lattice().omega() gp: if(E.disc>0,2,1)E.omega[1] magma: (Discriminant(E) gt 0 select 2 else 1) RealPeriod(E);
Tamagawa product:	$\prod_{p}c_p$	=	$6$ = $1\cdot1\cdot3\cdot2\cdot1$	comment: Tamagawa numbers sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] \| i<-[1..#gr[4][,1]]] magma: TamagawaNumbers(E); oscar: tamagawa_numbers(E)
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$	comment: Torsion order sage: E.torsion_order() gp: elltors(E)[1] magma: Order(TorsionSubgroup(E)); oscar: prod(torsion_structure(E)[1])
Special value:	$L'(E,1)$	≈	$3.9376336491734448381779941949$	comment: Special L-value r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial() gp: [r,L1r] = ellanalyticrank(E); L1r/r! magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
Analytic order of Ш:	Ш ${}_{\mathrm{an}}$	≈	$1$ (rounded)	comment: Order of Sha sage: E.sha().an_numerical() magma: MordellWeilShaInformation(E);

$\displaystyle 3.937633649 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.049919 \cdot 13.146770 \cdot 6}{1^2} \approx 3.937633649$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); 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)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */

E := EllipticCurve(%s); 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 438702.2.a.f

q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} + 3 q^{7} - q^{8} + q^{9} + 2 q^{10} + q^{11} - q^{12} - 3 q^{13} - 3 q^{14} + 2 q^{15} + q^{16} - q^{18} + O(q^{20})

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ 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

magma: ModularForm(E);

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

Modular degree:	29859840	comment: Modular degree sage: E.modular_degree() gp: ellmoddegree(E) magma: ModularDegree(E);
$\Gamma_0(N)$ -optimal:	yes
Manin constant:	1	comment: Manin constant magma: ManinConstant(E);

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$1$	$I_{9}$	nonsplit multiplicative	1	1	9	9
$3$	$1$	$I_{3}$	nonsplit multiplicative	1	1	3	3
$11$	$3$	$I_{3}$	split multiplicative	-1	1	3	3
$17$	$2$	$I_{4}^{*}$	additive	1	2	10	4
$23$	$1$	$I_{1}$	split multiplicative	-1	1	1	1

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

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 $\ell$ .

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[1519, 2, 0, 1], [4049, 2, 4049, 3], [1, 0, 2, 1], [1, 2, 0, 1], [3961, 2, 3961, 3], [6071, 2, 6070, 3], [3313, 2, 3313, 3], [3037, 2, 3037, 3], [1, 1, 6071, 0]]

GL(2,Integers(6072)).subgroup(gens)

Gens := [[1519, 2, 0, 1], [4049, 2, 4049, 3], [1, 0, 2, 1], [1, 2, 0, 1], [3961, 2, 3961, 3], [6071, 2, 6070, 3], [3313, 2, 3313, 3], [3037, 2, 3037, 3], [1, 1, 6071, 0]];

sub<GL(2,Integers(6072))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $6072 = 2^{3} \cdot 3 \cdot 11 \cdot 23$ , index $2$ , genus $0$ , and generators

$\left(\begin{array}{rr} 1519 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4049 & 2 \\ 4049 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3961 & 2 \\ 3961 & 3 \end{array}\right),\left(\begin{array}{rr} 6071 & 2 \\ 6070 & 3 \end{array}\right),\left(\begin{array}{rr} 3313 & 2 \\ 3313 & 3 \end{array}\right),\left(\begin{array}{rr} 3037 & 2 \\ 3037 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 6071 & 0 \end{array}\right)$ .

The torsion field $K:=\Q(E[6072])$ is a degree- $130005231206400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6072\Z)$ .

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
$2$	nonsplit multiplicative	$4$	$219351 = 3 \cdot 11 \cdot 17^{2} \cdot 23$
$3$	nonsplit multiplicative	$4$	$6647 = 17^{2} \cdot 23$
$11$	split multiplicative	$12$	$39882 = 2 \cdot 3 \cdot 17^{2} \cdot 23$
$17$	additive	$162$	$1518 = 2 \cdot 3 \cdot 11 \cdot 23$
$23$	split multiplicative	$24$	$19074 = 2 \cdot 3 \cdot 11 \cdot 17^{2}$

$p$ -adic regulators

$p$ -adic regulators are not yet computed for curves that are not $\Gamma_0$ -optimal.

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

Mordell-Weil group structure

Mordell-Weil generators

Integral points

Invariants

BSD invariants

BSD formula

Modular invariants

Local data at primes of bad reduction

Galois representations

Isogenies

Twists

Iwasawa invariants

$p$ -adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	438702f1

Conductor	$438702$
Discriminant	$-8.532\times 10^{20}$
j-invariant	$-\frac{629989223007953593}{35345595428352}$
CM	no
Rank	$1$
Torsion structure	trivial

Properties

Related objects

Downloads

Learn more

Mordell-Weil group structure

Invariants

Local data at primes of bad reduction

ppp-adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

$p$ -adic regulators