LMFDB - Elliptic curve with Cremona label 438702b2 (LMFDB label 438702.b1)

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

$y^2+xy=x^3+x^2-125304x-17124696$ y^2+xy=x^3+x^2-125304x-17124696	(homogenize, simplify)
$y^2z+xyz=x^3+x^2z-125304xz^2-17124696z^3$ y^2z+xyz=x^3+x^2z-125304xz^2-17124696z^3	(dehomogenize, simplify)
$y^2=x^3-162394659x-796533900066$ y^2=x^3-162394659x-796533900066	(homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 1, 0, -125304, -17124696])

gp: E = ellinit([1, 1, 0, -125304, -17124696])

magma: E := EllipticCurve([1, 1, 0, -125304, -17124696]);

oscar: E = elliptic_curve([1, 1, 0, -125304, -17124696])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Mordell-Weil group structure

$\Z \oplus \Z$

magma: MordellWeilGroup(E);

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(-205, 97)$	$0.96598995884768963220256501439$	$\infty$
$(-10023/49, 36714/343)$	$3.1091903253455793493775583500$	$\infty$

$\left(-205, 108\right)$ , $\left(-205, 97\right)$ , $\left(411, 801\right)$ , $\left(411, -1212\right)$ , $\left(961, 26926\right)$ , $\left(961, -27887\right)$ , $\left(3109803, 5482469793\right)$ , $\left(3109803, -5485579596\right)$ (-205, 108), (-205, 97), (411, 801), (411, -1212), (961, 26926), (961, -27887), (3109803, 5482469793), (3109803, -5485579596)

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:	$\Delta$	=	$847840749624$	=	$2^{3} \cdot 3^{2} \cdot 11^{6} \cdot 17^{2} \cdot 23$	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	$j$	=	$\frac{752880570738483817}{2933705016}$	=	$2^{-3} \cdot 3^{-2} \cdot 7^{3} \cdot 11^{-6} \cdot 17 \cdot 23^{-1} \cdot 50543^{3}$	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:	$h_{\mathrm{Faltings}}$	≈	$1.5018012013543317265947685281$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$1.0295989773449623798865127585$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$Q$	≈	$0.9459165128632454$
Szpiro ratio:	$\sigma_{m}$	≈	$3.6045750699854615$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$2$	sage: E.analytic_rank() gp: ellanalyticrank(E) magma: AnalyticRank(E);
Mordell-Weil rank:	$r$	=	$2$	comment: Rank sage: E.rank() gp: [lower,upper] = ellrank(E) magma: Rank(E);
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$2.9934638393989108039178844997$	comment: Regulator sage: E.regulator() G = E.gen \\ if available matdet(ellheightmatrix(E,G)) magma: Regulator(E);
Real period:	$\Omega$	≈	$0.25375112198107688216894551741$	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:	$\prod_{p}c_p$	=	$12$ = $1\cdot2\cdot( 2 \cdot 3 )\cdot1\cdot1$	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:	$L^{(2)}(E,1)/2!$	≈	$9.1151376942870690373936661599$	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);

BSD formula

$\displaystyle 9.115137694 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.253751 \cdot 2.993464 \cdot 12}{1^2} \approx 9.115137694$

# 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.b

q - q^{2} - q^{3} + q^{4} - 3 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + 3 q^{10} + q^{11} - q^{12} + 2 q^{13} - q^{14} + 3 q^{15} + q^{16} - q^{18} + 2 q^{19} + 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:	1990656	comment: Modular degree sage: E.modular_degree() gp: ellmoddegree(E) magma: ModularDegree(E);
$\Gamma_0(N)$ -optimal:	no
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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$1$	$I_{3}$	nonsplit multiplicative	1	1	3	3
$3$	$2$	$I_{2}$	nonsplit multiplicative	1	1	2	2
$11$	$6$	$I_{6}$	split multiplicative	-1	1	6	6
$17$	$1$	$II$	additive	1	2	2	0
$23$	$1$	$I_{1}$	nonsplit 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 $\ell$ except those listed in the table below.

prime $\ell$	mod- $\ell$ image	$\ell$ -adic image
$3$	3B	3.4.0.1

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 = [[2763, 2, 6082, 7], [3, 4, 8, 11], [4, 3, 9, 7], [1, 0, 6, 1], [9379, 6, 9378, 7], [2857, 6, 8571, 19], [4693, 6, 4695, 19], [7039, 6, 2349, 19], [6650, 2739, 1183, 4699], [1, 6, 0, 1]]

GL(2,Integers(9384)).subgroup(gens)

Gens := [[2763, 2, 6082, 7], [3, 4, 8, 11], [4, 3, 9, 7], [1, 0, 6, 1], [9379, 6, 9378, 7], [2857, 6, 8571, 19], [4693, 6, 4695, 19], [7039, 6, 2349, 19], [6650, 2739, 1183, 4699], [1, 6, 0, 1]];

sub<GL(2,Integers(9384))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $9384 = 2^{3} \cdot 3 \cdot 17 \cdot 23$ , index $16$ , genus $0$ , and generators

$\left(\begin{array}{rr} 2763 & 2 \\ 6082 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 9379 & 6 \\ 9378 & 7 \end{array}\right),\left(\begin{array}{rr} 2857 & 6 \\ 8571 & 19 \end{array}\right),\left(\begin{array}{rr} 4693 & 6 \\ 4695 & 19 \end{array}\right),\left(\begin{array}{rr} 7039 & 6 \\ 2349 & 19 \end{array}\right),\left(\begin{array}{rr} 6650 & 2739 \\ 1183 & 4699 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$ .

The torsion field $K:=\Q(E[9384])$ is a degree- $96440244240384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9384\Z)$ .

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	$4$	$6647 = 17^{2} \cdot 23$
$3$	nonsplit multiplicative	$4$	$6647 = 17^{2} \cdot 23$
$11$	split multiplicative	$12$	$39882 = 2 \cdot 3 \cdot 17^{2} \cdot 23$
$17$	additive	$66$	$1518 = 2 \cdot 3 \cdot 11 \cdot 23$
$23$	nonsplit 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	438702b2

Conductor	$438702$
Discriminant	$847840749624$
j-invariant	$\frac{752880570738483817}{2933705016}$
CM	no
Rank	$2$
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