# Properties

 Label 1045b4 Conductor $1045$ Discriminant $-1.532\times 10^{19}$ j-invariant $$-\frac{94256762600623910012361}{15323275604248046875}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3-x^2-948119x-401927292$$ y^2+xy=x^3-x^2-948119x-401927292 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-948119xz^2-401927292z^3$$ y^2z+xyz=x^3-x^2z-948119xz^2-401927292z^3 (dehomogenize, simplify) $$y^2=x^3-15169907x-25738516594$$ y^2=x^3-15169907x-25738516594 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 0, -948119, -401927292])

gp: E = ellinit([1, -1, 0, -948119, -401927292])

magma: E := EllipticCurve([1, -1, 0, -948119, -401927292]);

oscar: E = EllipticCurve([1, -1, 0, -948119, -401927292])

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

 $P$ = $$\left(1292, 22344\right)$$ (1292, 22344) $\hat{h}(P)$ ≈ $1.6638179689958272347419915734$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(\frac{4563}{4}, -\frac{4563}{8}\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(1292, 22344\right)$$, $$\left(1292, -23636\right)$$, $$\left(20672, 2958414\right)$$, $$\left(20672, -2979086\right)$$

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: $-15323275604248046875$ = $-1 \cdot 5^{16} \cdot 11^{4} \cdot 19^{3}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{94256762600623910012361}{15323275604248046875}$$ = $-1 \cdot 3^{3} \cdot 5^{-16} \cdot 11^{-4} \cdot 19^{-3} \cdot 61^{3} \cdot 431^{3} \cdot 577^{3}$ 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: $2.4089712693847815425058418295\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $2.4089712693847815425058418295\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1.6638179689958272347419915734\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.075822212874574098349463719319\dots$ 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: $96$  = $2^{4}\cdot2\cdot3$ 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: $L'(E,1)$ ≈ $3.0277046455138353542434232564$ 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

$\displaystyle 3.027704646 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.075822 \cdot 1.663818 \cdot 96}{2^2} \approx 3.027704646$

# 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("anayltic 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

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

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: 17664
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 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$5$ $16$ $I_{16}$ Split multiplicative -1 1 16 16
$11$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$19$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

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 4.12.0.8

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 = [[761, 8, 1372, 33], [624, 3, 181, 2], [1665, 8, 1664, 9], [1, 0, 8, 1], [635, 628, 674, 1469], [7, 6, 1666, 1667], [1467, 1466, 1058, 219], [1, 8, 0, 1], [1, 4, 4, 17]]

GL(2,Integers(1672)).subgroup(gens)

Gens := [[761, 8, 1372, 33], [624, 3, 181, 2], [1665, 8, 1664, 9], [1, 0, 8, 1], [635, 628, 674, 1469], [7, 6, 1666, 1667], [1467, 1466, 1058, 219], [1, 8, 0, 1], [1, 4, 4, 17]];

sub<GL(2,Integers(1672))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$1672 = 2^{3} \cdot 11 \cdot 19$$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 761 & 8 \\ 1372 & 33 \end{array}\right),\left(\begin{array}{rr} 624 & 3 \\ 181 & 2 \end{array}\right),\left(\begin{array}{rr} 1665 & 8 \\ 1664 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 635 & 628 \\ 674 & 1469 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1666 & 1667 \end{array}\right),\left(\begin{array}{rr} 1467 & 1466 \\ 1058 & 219 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.

The torsion field $K:=\Q(E[1672])$ is a degree-$4815382$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1672\Z)$.

## $p$-adic regulators

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

## 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 ss split ss nonsplit ord ord split ord ord ord ord ord ord ss 1 3,3 2 3,1 1 3 1 2 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

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 1045b consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $K$ $2$ $$\Q(\sqrt{-19})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{19})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ Not in database $4$ 4.0.109744.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{19})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.192699928576.2 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.4.45141114468499456.5 $$\Z/8\Z$$ Not in database $8$ 8.0.346384039936.3 $$\Z/8\Z$$ Not in database $8$ 8.2.20012416875.1 $$\Z/6\Z$$ 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 $16$ 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.