Properties

 Label 4650bd2 Conductor $4650$ Discriminant $4.982\times 10^{13}$ j-invariant $$\frac{1152829477932246539641}{3188367360}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

 $$y^2+xy+y=x^3+x^2-5461188x+4909955781$$ y^2+xy+y=x^3+x^2-5461188x+4909955781 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3+x^2z-5461188xz^2+4909955781z^3$$ y^2z+xyz+yz^2=x^3+x^2z-5461188xz^2+4909955781z^3 (dehomogenize, simplify) $$y^2=x^3-7077699675x+229185062421750$$ y^2=x^3-7077699675x+229185062421750 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 1, 1, -5461188, 4909955781])

gp: E = ellinit([1, 1, 1, -5461188, 4909955781])

magma: E := EllipticCurve([1, 1, 1, -5461188, 4909955781]);

oscar: E = elliptic_curve([1, 1, 1, -5461188, 4909955781])

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);

Infinite order Mordell-Weil generator and height

 $P$ = $$\left(1405, 2897\right)$$ (1405, 2897) $\hat{h}(P)$ ≈ $0.39612412703278941597770964800$

sage: E.gens()

magma: Generators(E);

gp: E.gen

Torsion generators

$$\left(\frac{5395}{4}, -\frac{5399}{8}\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

Integral points

$$\left(-395, 83897\right)$$, $$\left(-395, -83503\right)$$, $$\left(969, 22517\right)$$, $$\left(969, -23487\right)$$, $$\left(1341, -175\right)$$, $$\left(1341, -1167\right)$$, $$\left(1349, -663\right)$$, $$\left(1349, -687\right)$$, $$\left(1405, 2897\right)$$, $$\left(1405, -4303\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

Invariants

 Conductor: $$4650$$ = $2 \cdot 3 \cdot 5^{2} \cdot 31$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $49818240000000$ = $2^{13} \cdot 3^{4} \cdot 5^{7} \cdot 31^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{1152829477932246539641}{3188367360}$$ = $2^{-13} \cdot 3^{-4} \cdot 5^{-1} \cdot 17^{3} \cdot 31^{-2} \cdot 616793^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.2864422954146134653708761084\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $1.4817233391975632780704964418\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0264041155005181\dots$ Szpiro ratio: $6.8864101116835315\dots$

BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.39612412703278941597770964800\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.41831522806529736356304299586\dots$ 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: $104$  = $13\cdot2\cdot2\cdot2$ 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: $2$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ ( rounded) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $4.3083236180890913581510438563$ 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);

BSD formula

$\displaystyle 4.308323618 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.418315 \cdot 0.396124 \cdot 104}{2^2} \approx 4.308323618$

# 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 analyiic 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

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

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);

Modular degree: 99840
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

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

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $13$ $I_{13}$ split multiplicative -1 1 13 13
$3$ $2$ $I_{4}$ nonsplit multiplicative 1 1 4 4
$5$ $2$ $I_{1}^{*}$ additive 1 2 7 1
$31$ $2$ $I_{2}$ split multiplicative -1 1 2 2

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 2.3.0.1

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 = [[2, 1, 619, 0], [1, 0, 4, 1], [3, 4, 8, 11], [1, 2, 2, 5], [1, 4, 0, 1], [777, 466, 464, 775], [994, 1, 743, 0], [1237, 4, 1236, 5], [561, 4, 1122, 9]]

GL(2,Integers(1240)).subgroup(gens)

Gens := [[2, 1, 619, 0], [1, 0, 4, 1], [3, 4, 8, 11], [1, 2, 2, 5], [1, 4, 0, 1], [777, 466, 464, 775], [994, 1, 743, 0], [1237, 4, 1236, 5], [561, 4, 1122, 9]];

sub<GL(2,Integers(1240))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$1240 = 2^{3} \cdot 5 \cdot 31$$, index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 2 & 1 \\ 619 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 777 & 466 \\ 464 & 775 \end{array}\right),\left(\begin{array}{rr} 994 & 1 \\ 743 & 0 \end{array}\right),\left(\begin{array}{rr} 1237 & 4 \\ 1236 & 5 \end{array}\right),\left(\begin{array}{rr} 561 & 4 \\ 1122 & 9 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[1240])$ is a degree-$54853632000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1240\Z)$.

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
$2$ split multiplicative $4$ $$25 = 5^{2}$$
$3$ nonsplit multiplicative $4$ $$1550 = 2 \cdot 5^{2} \cdot 31$$
$5$ additive $18$ $$186 = 2 \cdot 3 \cdot 31$$
$13$ good $2$ $$2325 = 3 \cdot 5^{2} \cdot 31$$
$31$ split multiplicative $32$ $$150 = 2 \cdot 3 \cdot 5^{2}$$

Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 4650bd consists of 2 curves linked by isogenies of degree 2.

Twists

The minimal quadratic twist of this elliptic curve is 930j2, its twist by $5$.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{10})$$ $$\Z/2\Z \oplus \Z/2\Z$$ not in database $4$ 4.0.153760.1 $$\Z/4\Z$$ not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/4\Z$$ not in database $8$ 8.0.37827420160000.5 $$\Z/2\Z \oplus \Z/4\Z$$ not in database $8$ deg 8 $$\Z/6\Z$$ not in database $16$ deg 16 $$\Z/8\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 split nonsplit add ord ord ord ss ss ord ss split ord ord ord ord 9 1 - 1 1 1 1,1 1,1 1 1,1 2 1 1 1 1 0 0 - 0 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.

$p$-adic regulators

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