# Properties

 Label 1045.a2 Conductor $1045$ Discriminant $5225$ j-invariant $$\frac{24137569}{5225}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3-6x-5$$ y^2+xy=x^3-6x-5 (homogenize, simplify) $$y^2z+xyz=x^3-6xz^2-5z^3$$ y^2z+xyz=x^3-6xz^2-5z^3 (dehomogenize, simplify) $$y^2=x^3-7803x-209898$$ y^2=x^3-7803x-209898 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 0, -6, -5])

gp: E = ellinit([1, 0, 0, -6, -5])

magma: E := EllipticCurve([1, 0, 0, -6, -5]);

oscar: E = EllipticCurve([1, 0, 0, -6, -5])

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

 $P$ = $$\left(3, 1\right)$$ (3, 1) $\hat{h}(P)$ ≈ $0.68430808120701749338340969163$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(-2, 1\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-2, 1\right)$$, $$\left(-1, 1\right)$$, $$\left(-1, 0\right)$$, $$\left(3, 1\right)$$, $$\left(3, -4\right)$$, $$\left(14, 45\right)$$, $$\left(14, -59\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$1045$$ = $5 \cdot 11 \cdot 19$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $5225$ = $5^{2} \cdot 11 \cdot 19$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{24137569}{5225}$$ = $5^{-2} \cdot 11^{-1} \cdot 17^{6} \cdot 19^{-1}$ 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: $-0.57261717817892023429300942008\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.57261717817892023429300942008\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.68430808120701749338340969163\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $3.0946515238652600647936815354\dots$ 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: $2$  = $2\cdot1\cdot1$ 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: $1$ (exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $1.0588475231503044094353175337$ 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

$\displaystyle 1.058847523 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.094652 \cdot 0.684308 \cdot 2}{2^2} \approx 1.058847523$

# 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

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

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: 64
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

This elliptic curve is semistable. There are 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$11$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$19$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 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 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 = [[1, 0, 4, 1], [3137, 1046, 1044, 3135], [3, 4, 8, 11], [1522, 1, 3419, 0], [1, 2, 2, 5], [1, 4, 0, 1], [3082, 1, 2639, 0], [4177, 4, 4176, 5], [837, 4, 1674, 9]]

GL(2,Integers(4180)).subgroup(gens)

Gens := [[1, 0, 4, 1], [3137, 1046, 1044, 3135], [3, 4, 8, 11], [1522, 1, 3419, 0], [1, 2, 2, 5], [1, 4, 0, 1], [3082, 1, 2639, 0], [4177, 4, 4176, 5], [837, 4, 1674, 9]];

sub<GL(2,Integers(4180))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$4180 = 2^{2} \cdot 5 \cdot 11 \cdot 19$$, index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3137 & 1046 \\ 1044 & 3135 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1522 & 1 \\ 3419 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3082 & 1 \\ 2639 & 0 \end{array}\right),\left(\begin{array}{rr} 4177 & 4 \\ 4176 & 5 \end{array}\right),\left(\begin{array}{rr} 837 & 4 \\ 1674 & 9 \end{array}\right)$.

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

The torsion field $K:=\Q(E[4180])$ is a degree-$6240706560000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4180\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 1045.a consists of 2 curves linked by isogenies of degree 2.

## 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 $K$ $2$ $$\Q(\sqrt{209})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.5225.1 $$\Z/4\Z$$ Not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.1192518600625.3 $$\Z/2\Z \oplus \Z/4\Z$$ 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 ord ord nonsplit ord nonsplit ord ss nonsplit ss ord ord ss ord ord ord 1 1 1 1 1 1 1,5 1 1,1 1 1 1,1 1 1 1 0 0 0 0 0 0 0,0 0 0,0 0 0 0,0 0 0 0

## $p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.