LMFDB - Elliptic curve with Cremona label 485184fq1 (LMFDB label 485184.fq3)

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Simplified equation

$y^2=x^3+x^2-1346289x+600797967$ y^2=x^3+x^2-1346289x+600797967	(homogenize, simplify)
$y^2z=x^3+x^2z-1346289xz^2+600797967z^3$ y^2z=x^3+x^2z-1346289xz^2+600797967z^3	(dehomogenize, simplify)
$y^2=x^3-109049436x+438308866224$ y^2=x^3-109049436x+438308866224	(homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 1, 0, -1346289, 600797967])

gp: E = ellinit([0, 1, 0, -1346289, 600797967])

magma: E := EllipticCurve([0, 1, 0, -1346289, 600797967]);

oscar: E = elliptic_curve([0, 1, 0, -1346289, 600797967])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Mordell-Weil group structure

$\Z \oplus \Z/{2}\Z$

magma: MordellWeilGroup(E);

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(-697, 34656)$	$2.2413992536627798611351934736$	$\infty$
$(671, 0)$	$0$	$2$

Integral points

$(-697,\pm 34656)$ , $\left(671, 0\right)$ (-697, 34656), (-697, -34656), (671, 0)

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

Invariants

Conductor:	$N$	=	$485184$	=	$2^{6} \cdot 3 \cdot 7 \cdot 19^{2}$	comment: Conductor sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); oscar: conductor(E)
Discriminant:	$\Delta$	=	$2152843602051072$	=	$2^{14} \cdot 3 \cdot 7^{2} \cdot 19^{7}$	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	$j$	=	$\frac{350104249168}{2793}$	=	$2^{4} \cdot 3^{-1} \cdot 7^{-2} \cdot 19^{-1} \cdot 2797^{3}$	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:	$h_{\mathrm{Faltings}}$	≈	$2.1142444924048072515798594693$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.16664670783168250607809172168$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$Q$	≈	$0.8976750426615946$
Szpiro ratio:	$\sigma_{m}$	≈	$4.120915292546831$

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:	$\mathrm{Reg}(E/\Q)$	≈	$2.2413992536627798611351934736$	comment: Regulator sage: E.regulator() G = E.gen \\ if available matdet(ellheightmatrix(E,G)) magma: Regulator(E);
Real period:	$\Omega$	≈	$0.41594451060140488006911240807$	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:	$\prod_{p}c_p$	=	$32$ = $2^{2}\cdot1\cdot2\cdot2^{2}$	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:	$\#E(\Q)_{\mathrm{tor}}$	=	$2$	comment: Torsion order sage: E.torsion_order() gp: elltors(E)[1] magma: Order(TorsionSubgroup(E)); oscar: prod(torsion_structure(E)[1])
Special value:	$L'(E,1)$	≈	$7.4583817250169529913104662067$	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);

$\displaystyle 7.458381725 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.415945 \cdot 2.241399 \cdot 32}{2^2} \approx 7.458381725$

# 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 485184.2.a.fq

q + q^{3} - 2 q^{5} + q^{7} + q^{9} - 4 q^{11} - 6 q^{13} - 2 q^{15} + 2 q^{17} + O(q^{20})

q + q^3 - 2*q^5 + q^7 + q^9 - 4*q^11 - 6*q^13 - 2*q^15 + 2*q^17 + 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:	7372800	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 4 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$4$	$I_{4}^{*}$	additive	-1	6	14	0
$3$	$1$	$I_{1}$	split multiplicative	-1	1	1	1
$7$	$2$	$I_{2}$	split multiplicative	-1	1	2	2
$19$	$4$	$I_{1}^{*}$	additive	-1	2	7	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
$2$	2B	8.12.0.11

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 = [[449, 8, 448, 9], [1, 0, 8, 1], [277, 282, 166, 397], [1, 8, 0, 1], [1, 4, 4, 17], [7, 6, 450, 451], [308, 1, 175, 6], [68, 455, 49, 450], [167, 168, 386, 161]]

GL(2,Integers(456)).subgroup(gens)

Gens := [[449, 8, 448, 9], [1, 0, 8, 1], [277, 282, 166, 397], [1, 8, 0, 1], [1, 4, 4, 17], [7, 6, 450, 451], [308, 1, 175, 6], [68, 455, 49, 450], [167, 168, 386, 161]];

sub<GL(2,Integers(456))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $456 = 2^{3} \cdot 3 \cdot 19$ , index $48$ , genus $0$ , and generators

$\left(\begin{array}{rr} 449 & 8 \\ 448 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 277 & 282 \\ 166 & 397 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 450 & 451 \end{array}\right),\left(\begin{array}{rr} 308 & 1 \\ 175 & 6 \end{array}\right),\left(\begin{array}{rr} 68 & 455 \\ 49 & 450 \end{array}\right),\left(\begin{array}{rr} 167 & 168 \\ 386 & 161 \end{array}\right)$ .

The torsion field $K:=\Q(E[456])$ is a degree- $189112320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/456\Z)$ .

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
$2$	additive	$2$	$1083 = 3 \cdot 19^{2}$
$3$	split multiplicative	$4$	$161728 = 2^{6} \cdot 7 \cdot 19^{2}$
$7$	split multiplicative	$8$	$69312 = 2^{6} \cdot 3 \cdot 19^{2}$
$19$	additive	$200$	$1344 = 2^{6} \cdot 3 \cdot 7$

$p$ -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	485184fq1

Conductor	$485184$
Discriminant	$2.153\times 10^{15}$
j-invariant	$\frac{350104249168}{2793}$
CM	no
Rank	$1$
Torsion structure	$\Z/{2}\Z$

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