# Properties

 Label 768.b1 Conductor $768$ Discriminant $4608$ j-invariant $$\frac{2744000}{9}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3-x^2-23x+51$$ y^2=x^3-x^2-23x+51 (homogenize, simplify) $$y^2z=x^3-x^2z-23xz^2+51z^3$$ y^2z=x^3-x^2z-23xz^2+51z^3 (dehomogenize, simplify) $$y^2=x^3-1890x+31536$$ y^2=x^3-1890x+31536 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, -1, 0, -23, 51])

gp: E = ellinit([0, -1, 0, -23, 51])

magma: E := EllipticCurve([0, -1, 0, -23, 51]);

oscar: E = elliptic_curve([0, -1, 0, -23, 51])

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

 $P$ = $$\left(2, 3\right)$$ (2, 3) $\hat{h}(P)$ ≈ $0.43929586130482440834533239014$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(3, 0\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$(-5,\pm 4)$$, $$(2,\pm 3)$$, $$\left(3, 0\right)$$, $$(5,\pm 6)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$768$$ = $2^{8} \cdot 3$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $4608$ = $2^{9} \cdot 3^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{2744000}{9}$$ = $2^{6} \cdot 3^{-2} \cdot 5^{3} \cdot 7^{3}$ 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: $-0.43171442438310503261991559624\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.95157480980306401468283968733\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.106715441314673\dots$ Szpiro ratio: $3.1703670252803358\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.43929586130482440834533239014\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $4.3656747145718101333424083404\dots$ 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: $4$  = $2\cdot2$ 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: $L'(E,1)$ ≈ $1.9178228339145167907527079609$ 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

$\displaystyle 1.917822834 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 4.365675 \cdot 0.439296 \cdot 4}{2^2} \approx 1.917822834$

# 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

$$q - q^{3} - 4 q^{7} + q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})$$

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: 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 not semistable. There are 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $III$ Additive 1 8 9 0
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

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 16.48.0.247

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, 16, 1], [1, 16, 0, 1], [11, 8, 16, 3], [17, 10, 46, 45], [34, 13, 27, 10], [13, 8, 8, 5], [11, 12, 44, 35], [33, 16, 32, 17], [41, 38, 10, 3]]

GL(2,Integers(48)).subgroup(gens)

Gens := [[1, 0, 16, 1], [1, 16, 0, 1], [11, 8, 16, 3], [17, 10, 46, 45], [34, 13, 27, 10], [13, 8, 8, 5], [11, 12, 44, 35], [33, 16, 32, 17], [41, 38, 10, 3]];

sub<GL(2,Integers(48))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 48.96.1-24.dl.1.3, level $$48 = 2^{4} \cdot 3$$, index $96$, genus $1$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 8 \\ 16 & 3 \end{array}\right),\left(\begin{array}{rr} 17 & 10 \\ 46 & 45 \end{array}\right),\left(\begin{array}{rr} 34 & 13 \\ 27 & 10 \end{array}\right),\left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 11 & 12 \\ 44 & 35 \end{array}\right),\left(\begin{array}{rr} 33 & 16 \\ 32 & 17 \end{array}\right),\left(\begin{array}{rr} 41 & 38 \\ 10 & 3 \end{array}\right)$.

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

The torsion field $K:=\Q(E[48])$ is a degree-$12288$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 768.b 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 $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.2048.1 $$\Z/4\Z$$ Not in database $8$ 8.4.1358954496.2 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.16777216.2 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.603979776.3 $$\Z/8\Z$$ Not in database $8$ 8.0.603979776.2 $$\Z/8\Z$$ Not in database $8$ 8.2.2972033482752.16 $$\Z/6\Z$$ Not in database $16$ 16.0.1846757322198614016.5 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ 16.0.1459166279268040704.1 $$\Z/2\Z \oplus \Z/8\Z$$ 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 add nonsplit ss ord ord ord ord ord ord ord ord ord ord ord ord - 3 1,1 1 1 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

An entry - indicates that the invariants are not computed because the reduction is additive.

## $p$-adic regulators

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