# Properties

 Label 16245.c3 Conductor $16245$ Discriminant $-7.382\times 10^{18}$ j-invariant $$-\frac{147281603041}{215233605}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2-357458x-154357954$$ y^2+xy+y=x^3-x^2-357458x-154357954 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-357458xz^2-154357954z^3$$ y^2z+xyz+yz^2=x^3-x^2z-357458xz^2-154357954z^3 (dehomogenize, simplify) $$y^2=x^3-5719323x-9884628362$$ y^2=x^3-5719323x-9884628362 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 1, -357458, -154357954])

gp: E = ellinit([1, -1, 1, -357458, -154357954])

magma: E := EllipticCurve([1, -1, 1, -357458, -154357954]);

oscar: E = EllipticCurve([1, -1, 1, -357458, -154357954])

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

 $P$ = $$\left(\frac{257475799}{14884}, \frac{4113143706683}{1815848}\right)$$ (257475799/14884, 4113143706683/1815848) $\hat{h}(P)$ ≈ $16.444300196502978986772768780$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(\frac{3003}{4}, -\frac{3007}{8}\right)$$

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: $$16245$$ = $3^{2} \cdot 5 \cdot 19^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-7381747980094602645$ = $-1 \cdot 3^{22} \cdot 5 \cdot 19^{6}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{147281603041}{215233605}$$ = $-1 \cdot 3^{-16} \cdot 5^{-1} \cdot 5281^{3}$ 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: $2.3123952423943765976757719587\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.29086960847710152197363562429\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0594919023465201\dots$ Szpiro ratio: $5.284271984479717\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $16.444300196502978986772768780\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.092757202448663634451023939675\dots$ 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: $8$  = $2^{2}\cdot1\cdot2$ 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: $L'(E,1)$ ≈ $3.0506545649072520152891014913$ 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

$\displaystyle 3.050654565 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.092757 \cdot 16.444300 \cdot 8}{2^2} \approx 3.050654565$

# 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

$$q - q^{2} - q^{4} - q^{5} + 3 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} - q^{16} - 2 q^{17} + O(q^{20})$$

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);

Modular degree: 221184
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 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $4$ $I_{16}^{*}$ Additive -1 2 22 16
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$19$ $2$ $I_0^{*}$ Additive -1 2 6 0

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 16.48.0.134

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 = [[23, 18, 6558, 7115], [1, 0, 32, 1], [1, 32, 0, 1], [5, 28, 68, 381], [5340, 4351, 6479, 324], [5701, 2432, 2356, 2433], [7199, 0, 0, 9119], [9089, 32, 9088, 33], [1654, 3363, 1805, 7468], [6079, 6688, 304, 6687]]

GL(2,Integers(9120)).subgroup(gens)

Gens := [[23, 18, 6558, 7115], [1, 0, 32, 1], [1, 32, 0, 1], [5, 28, 68, 381], [5340, 4351, 6479, 324], [5701, 2432, 2356, 2433], [7199, 0, 0, 9119], [9089, 32, 9088, 33], [1654, 3363, 1805, 7468], [6079, 6688, 304, 6687]];

sub<GL(2,Integers(9120))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$9120 = 2^{5} \cdot 3 \cdot 5 \cdot 19$$, index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 23 & 18 \\ 6558 & 7115 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 5340 & 4351 \\ 6479 & 324 \end{array}\right),\left(\begin{array}{rr} 5701 & 2432 \\ 2356 & 2433 \end{array}\right),\left(\begin{array}{rr} 7199 & 0 \\ 0 & 9119 \end{array}\right),\left(\begin{array}{rr} 9089 & 32 \\ 9088 & 33 \end{array}\right),\left(\begin{array}{rr} 1654 & 3363 \\ 1805 & 7468 \end{array}\right),\left(\begin{array}{rr} 6079 & 6688 \\ 304 & 6687 \end{array}\right)$.

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

The torsion field $K:=\Q(E[9120])$ is a degree-$1452382617600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9120\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4, 8 and 16.
Its isogeny class 16245.c consists of 8 curves linked by isogenies of degrees dividing 16.

## Twists

The minimal quadratic twist of this elliptic curve is 15.a3, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-5})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{285})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-57})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-5}, \sqrt{-57})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{-57})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{10}, \sqrt{-57})$$ $$\Z/8\Z$$ Not in database $8$ 8.0.10809345024000000.207 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.4.42224004000000.33 $$\Z/8\Z$$ Not in database $8$ 8.0.432373800960000.125 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.2767192326144000000.13 $$\Z/16\Z$$ Not in database $8$ 8.0.2767192326144000000.19 $$\Z/16\Z$$ Not in database $8$ 8.2.14428733866875.1 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/16\Z$$ Not in database $16$ deg 16 $$\Z/16\Z$$ Not in database $16$ 16.0.7657353369870241665908736000000000000.34 $$\Z/2\Z \oplus \Z/16\Z$$ 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 11 13 17 19 23 29 31 37 41 43 47 ord add nonsplit ss ord ord ord add ss ord ss ord ord ord ord 3 - 3 1,5 1 1 1 - 1,3 1 1,1 1 1 1 1 3 - 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.