# Properties

 Label 7728.n1 Conductor $7728$ Discriminant $-3926071296$ j-invariant $$-\frac{3261064466}{1917027}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3+x^2-392x+4116$$ y^2=x^3+x^2-392x+4116 (homogenize, simplify) $$y^2z=x^3+x^2z-392xz^2+4116z^3$$ y^2z=x^3+x^2z-392xz^2+4116z^3 (dehomogenize, simplify) $$y^2=x^3-31779x+3095874$$ y^2=x^3-31779x+3095874 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -392, 4116])

gp: E = ellinit([0, 1, 0, -392, 4116])

magma: E := EllipticCurve([0, 1, 0, -392, 4116]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-14, 84\right)$$ (-14, 84) $\hat{h}(P)$ ≈ $0.054504924965534246102622620971$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-20,\pm 66)$$, $$(-14,\pm 84)$$, $$(7,\pm 42)$$, $$(10,\pm 36)$$, $$(28,\pm 126)$$, $$(42,\pm 252)$$, $$(658,\pm 16884)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$7728$$ = $2^{4} \cdot 3 \cdot 7 \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-3926071296$ = $-1 \cdot 2^{11} \cdot 3^{5} \cdot 7^{3} \cdot 23$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{3261064466}{1917027}$$ = $-1 \cdot 2 \cdot 3^{-5} \cdot 7^{-3} \cdot 11^{3} \cdot 23^{-1} \cdot 107^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.54346766786627462828385760656\dots$ Stable Faltings height: $-0.091917247647008572015271838110\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.054504924965534246102622620971\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $1.2914937554294764583891229884\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $60$  = $2^{2}\cdot5\cdot3\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L'(E,1)$ ≈ $4.2235662139883790731479646090$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q + q^{3} - 3 q^{5} + q^{7} + q^{9} - q^{13} - 3 q^{15} - 4 q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4800 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $4$ $I_{3}^{*}$ Additive 1 4 11 0
$3$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$23$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## Galois representations

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

The $\ell$-adic Galois representation has maximal image for all primes $\ell$.

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

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

## 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 split ord split ss ord ord ss nonsplit ord ord ord ord ord ord - 2 1 2 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 0

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

## Isogenies

This curve has no rational isogenies. Its isogeny class 7728.n consists of this curve only.

## 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 $K$ $3$ 3.1.3864.1 $$\Z/2\Z$$ Not in database $6$ 6.0.57691436544.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ 8.2.203050981896192.2 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.