# Properties

 Label 210.e3 Conductor $210$ Discriminant $-4.985\times 10^{15}$ j-invariant $$-\frac{187778242790732059201}{4984939585440150}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3-119300x-16229850$$ y^2+xy=x^3-119300x-16229850 (homogenize, simplify) $$y^2z+xyz=x^3-119300xz^2-16229850z^3$$ y^2z+xyz=x^3-119300xz^2-16229850z^3 (dehomogenize, simplify) $$y^2=x^3-154612827x-756756043146$$ y^2=x^3-154612827x-756756043146 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 0, -119300, -16229850])

gp: E = ellinit([1, 0, 0, -119300, -16229850])

magma: E := EllipticCurve([1, 0, 0, -119300, -16229850]);

oscar: E = elliptic_curve([1, 0, 0, -119300, -16229850])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(\frac{1599}{4}, -\frac{1599}{8}\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$210$$ = $2 \cdot 3 \cdot 5 \cdot 7$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-4984939585440150$ = $-1 \cdot 2 \cdot 3 \cdot 5^{2} \cdot 7^{16}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{187778242790732059201}{4984939585440150}$$ = $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{-16} \cdot 241^{3} \cdot 23761^{3}$ 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: $1.7944294168955630469530382474\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $1.7944294168955630469530382474\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0554678196515268\dots$ Szpiro ratio: $8.738676972435714\dots$

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.12824162625244165789596414857\dots$ 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: $4$  = $1\cdot1\cdot2\cdot2$ 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: $16$ = $4^2$ ( exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L(E,1)$ ≈ $2.0518660200390665263354263771$ 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

$\displaystyle 2.051866020 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{16 \cdot 0.128242 \cdot 1.000000 \cdot 4}{2^2} \approx 2.051866020$

# 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

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{14} + q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

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);

Modular degree: 2048
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 semistable. There are 4 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $1$ $I_{1}$ split multiplicative -1 1 1 1
$3$ $1$ $I_{1}$ split multiplicative -1 1 1 1
$5$ $2$ $I_{2}$ split multiplicative -1 1 2 2
$7$ $2$ $I_{16}$ nonsplit multiplicative 1 1 16 16

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 16.96.0.149

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 = [[1, 0, 32, 1], [1, 32, 0, 1], [30, 31, 2089, 3204], [2116, 29, 671, 770], [5, 28, 68, 381], [2711, 26, 2118, 875], [1471, 32, 2326, 513], [3329, 32, 3328, 33], [1921, 32, 496, 513], [23, 18, 798, 1355]]

GL(2,Integers(3360)).subgroup(gens)

Gens := [[1, 0, 32, 1], [1, 32, 0, 1], [30, 31, 2089, 3204], [2116, 29, 671, 770], [5, 28, 68, 381], [2711, 26, 2118, 875], [1471, 32, 2326, 513], [3329, 32, 3328, 33], [1921, 32, 496, 513], [23, 18, 798, 1355]];

sub<GL(2,Integers(3360))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7$$, index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 30 & 31 \\ 2089 & 3204 \end{array}\right),\left(\begin{array}{rr} 2116 & 29 \\ 671 & 770 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 2711 & 26 \\ 2118 & 875 \end{array}\right),\left(\begin{array}{rr} 1471 & 32 \\ 2326 & 513 \end{array}\right),\left(\begin{array}{rr} 3329 & 32 \\ 3328 & 33 \end{array}\right),\left(\begin{array}{rr} 1921 & 32 \\ 496 & 513 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 798 & 1355 \end{array}\right)$.

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

The torsion field $K:=\Q(E[3360])$ is a degree-$23781703680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3360\Z)$.

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 $4$ $$3$$
$3$ split multiplicative $4$ $$70 = 2 \cdot 5 \cdot 7$$
$5$ split multiplicative $6$ $$42 = 2 \cdot 3 \cdot 7$$
$7$ nonsplit multiplicative $8$ $$30 = 2 \cdot 3 \cdot 5$$

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4, 8 and 16.
Its isogeny class 210.e consists of 8 curves linked by isogenies of degrees dividing 16.

## Twists

This elliptic curve is its own minimal quadratic twist.

## 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 $K$ $2$ $$\Q(\sqrt{-6})$$ $$\Z/2\Z \oplus \Z/2\Z$$ not in database $2$ $$\Q(\sqrt{6})$$ $$\Z/4\Z$$ not in database $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ 2.0.4.1-22050.2-d1 $4$ $$\Q(i, \sqrt{6})$$ $$\Z/2\Z \oplus \Z/4\Z$$ not in database $4$ 4.0.1382400.3 $$\Z/2\Z \oplus \Z/4\Z$$ not in database $4$ 4.2.1382400.2 $$\Z/8\Z$$ not in database $4$ $$\Q(\zeta_{12})$$ $$\Z/8\Z$$ not in database $4$ $$\Q(\zeta_{8})$$ $$\Z/8\Z$$ not in database $8$ 8.0.7644119040000.32 $$\Z/4\Z \oplus \Z/4\Z$$ not in database $8$ 8.0.7644119040000.31 $$\Z/2\Z \oplus \Z/8\Z$$ not in database $8$ $$\Q(\zeta_{24})$$ $$\Z/2\Z \oplus \Z/8\Z$$ not in database $8$ 8.0.73383542784000000.51 $$\Z/2\Z \oplus \Z/8\Z$$ not in database $8$ 8.2.73383542784000000.14 $$\Z/16\Z$$ not in database $8$ 8.0.191102976.3 $$\Z/16\Z$$ not in database $8$ 8.0.849346560000.6 $$\Z/16\Z$$ not in database $8$ 8.0.10485760000.3 $$\Z/16\Z$$ not in database $8$ 8.2.4253299470000.8 $$\Z/6\Z$$ not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ not in database $16$ 16.0.9349208943630483456.10 $$\Z/2\Z \oplus \Z/16\Z$$ not in database $16$ 16.0.721389578983833600000000.8 $$\Z/2\Z \oplus \Z/16\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ not in database $16$ deg 16 $$\Z/12\Z$$ not in database $16$ deg 16 $$\Z/12\Z$$ 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 split split split nonsplit 2 5 1 0 3 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

## $p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.