# Properties

 Label 1922e2 Conductor $1922$ Discriminant $3844$ j-invariant $$\frac{51181724570498001}{4}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2-76332x+8136267$$ y^2+xy+y=x^3-x^2-76332x+8136267 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-76332xz^2+8136267z^3$$ y^2z+xyz+yz^2=x^3-x^2z-76332xz^2+8136267z^3 (dehomogenize, simplify) $$y^2=x^3-1221307x+519499798$$ y^2=x^3-1221307x+519499798 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 1, -76332, 8136267])

gp: E = ellinit([1, -1, 1, -76332, 8136267])

magma: E := EllipticCurve([1, -1, 1, -76332, 8136267]);

oscar: E = EllipticCurve([1, -1, 1, -76332, 8136267])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(\frac{639}{4}, -\frac{641}{8}\right)$$ (639/4, -641/8) $\hat{h}(P)$ ≈ $0.81588661340314983832465559114$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$1922$$ = $2 \cdot 31^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $3844$ = $2^{2} \cdot 31^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{51181724570498001}{4}$$ = $2^{-2} \cdot 3^{3} \cdot 31 \cdot 39397^{3}$ 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: $1.0529276699977781426504084647\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.48059646925025376832888107728\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.1080332911237138\dots$ Szpiro ratio: $5.996747016078877\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.81588661340314983832465559114\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $1.6954121794836243616166631538\dots$ 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: $2$  = $2\cdot1$ 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: $1$ 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: $L'(E,1)$ ≈ $2.7665282028826950308323632667$ 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

$\displaystyle 2.766528203 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.695412 \cdot 0.815887 \cdot 2}{1^2} \approx 2.766528203$

# 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

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

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: 5880
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$(j)_{-}$
$2$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$31$ $1$ $II$ Additive -1 2 2 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
$2$ 2Cn 2.2.0.1
$7$ 7B 7.8.0.1

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, 28, 1], [869, 28, 14, 393], [745, 28, 0, 1], [1303, 28, 882, 393], [1, 28, 0, 1], [1709, 28, 1708, 29], [1, 2, 14, 29], [15, 14, 1512, 1527], [698, 9, 357, 1627]]

GL(2,Integers(1736)).subgroup(gens)

Gens := [[1, 0, 28, 1], [869, 28, 14, 393], [745, 28, 0, 1], [1303, 28, 882, 393], [1, 28, 0, 1], [1709, 28, 1708, 29], [1, 2, 14, 29], [15, 14, 1512, 1527], [698, 9, 357, 1627]];

sub<GL(2,Integers(1736))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$1736 = 2^{3} \cdot 7 \cdot 31$$, index $576$, genus $16$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 28 & 1 \end{array}\right),\left(\begin{array}{rr} 869 & 28 \\ 14 & 393 \end{array}\right),\left(\begin{array}{rr} 745 & 28 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1303 & 28 \\ 882 & 393 \end{array}\right),\left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1709 & 28 \\ 1708 & 29 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 14 & 29 \end{array}\right),\left(\begin{array}{rr} 15 & 14 \\ 1512 & 1527 \end{array}\right),\left(\begin{array}{rr} 698 & 9 \\ 357 & 1627 \end{array}\right)$.

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

The torsion field $K:=\Q(E[1736])$ is a degree-$4799692800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1736\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 1922e consists of 2 curves linked by isogenies of degree 7.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.3.961.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ 6.6.481170140857.2 $$\Z/7\Z$$ Not in database $8$ 8.2.31055528805552.6 $$\Z/3\Z$$ Not in database $12$ 12.4.53715159415573774336.57 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $18$ 18.18.111402774653210244885538776688002793.1 $$\Z/2\Z \oplus \Z/14\Z$$ Not in database

We only show fields where the torsion growth is primitive.

## 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 split ss ord ord ord ord ord ord ord ord add ord ord ord ord 2 1,1 1 1 1 1 1 1 1 1 - 3 1 1 1 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.