# Properties

 Label 3249.g2 Conductor $3249$ Discriminant $1.114\times 10^{14}$ j-invariant $$\frac{30664297}{3249}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3-x^2-21186x+1078087$$ y^2+xy=x^3-x^2-21186x+1078087 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-21186xz^2+1078087z^3$$ y^2z+xyz=x^3-x^2z-21186xz^2+1078087z^3 (dehomogenize, simplify) $$y^2=x^3-338979x+68658590$$ y^2=x^3-338979x+68658590 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 0, -21186, 1078087])

gp: E = ellinit([1, -1, 0, -21186, 1078087])

magma: E := EllipticCurve([1, -1, 0, -21186, 1078087]);

oscar: E = EllipticCurve([1, -1, 0, -21186, 1078087])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(-\frac{34}{49}, \frac{358679}{343}\right)$$ (-34/49, 358679/343) $\hat{h}(P)$ ≈ $7.7855478445314400294318849727$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(-166, 83\right)$$, $$\left(62, -31\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-166, 83\right)$$, $$\left(62, -31\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$3249$$ = $3^{2} \cdot 19^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $111429157112001$ = $3^{8} \cdot 19^{8}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{30664297}{3249}$$ = $3^{-2} \cdot 19^{-2} \cdot 313^{3}$ 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: $1.4299433814344763273240879038\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.59158225248279874837804843061\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $0.9072668607940285\dots$ Szpiro ratio: $5.131881149859055\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $7.7855478445314400294318849727\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.57500719072589965602662337386\dots$ 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: $16$  = $2^{2}\cdot2^{2}$ 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: $4$ 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: $L'(E,1)$ ≈ $4.4767459943461067003013368241$ 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

$\displaystyle 4.476745994 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.575007 \cdot 7.785548 \cdot 16}{4^2} \approx 4.476745994$

# 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

$$q + q^{2} - q^{4} + 2 q^{5} - 3 q^{8} + 2 q^{10} - 6 q^{13} - q^{16} + 6 q^{17} + O(q^{20})$$

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);

Modular degree: 8640
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 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $4$ $I_{2}^{*}$ Additive -1 2 8 2
$19$ $4$ $I_{2}^{*}$ Additive -1 2 8 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
$2$ 2Cs 4.12.0.2

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 = [[117, 4, 2, 3], [225, 4, 224, 5], [165, 226, 194, 1], [1, 4, 0, 1], [151, 226, 0, 227], [1, 0, 4, 1]]

GL(2,Integers(228)).subgroup(gens)

Gens := [[117, 4, 2, 3], [225, 4, 224, 5], [165, 226, 194, 1], [1, 4, 0, 1], [151, 226, 0, 227], [1, 0, 4, 1]];

sub<GL(2,Integers(228))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$228 = 2^{2} \cdot 3 \cdot 19$$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 117 & 4 \\ 2 & 3 \end{array}\right),\left(\begin{array}{rr} 225 & 4 \\ 224 & 5 \end{array}\right),\left(\begin{array}{rr} 165 & 226 \\ 194 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 151 & 226 \\ 0 & 227 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[228])$ is a degree-$11819520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/228\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 3249.g consists of 4 curves linked by isogenies of degrees dividing 4.

## Twists

The minimal quadratic twist of this elliptic curve is 57.c2, its twist by $57$.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\sqrt{3}, \sqrt{19})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{19})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.2702336256.1 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.8334036681507.4 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ 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 ord add ord ss ss ord ord add ord ord ord ord ord ord ord ? - 1 1,1 1,1 1 1 - 1 1 1 1 1 1 1 ? - 0 0,0 0,0 0 0 - 0 0 0 0 0 0 0

An entry ? indicates that the invariants have not yet been computed.

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.