# Properties

 Label 48.a1 Conductor $48$ Discriminant $18432$ j-invariant $\frac{3065617154}{9}$ CM no Rank $0$ Torsion Structure $\Z/{4}\Z$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -384, 2772]); // or
magma: E := EllipticCurve("48a5");
sage: E = EllipticCurve([0, 1, 0, -384, 2772]) # or
sage: E = EllipticCurve("48a5")
gp: E = ellinit([0, 1, 0, -384, 2772]) \\ or
gp: E = ellinit("48a5")

$y^2 = x^{3} + x^{2} - 384 x + 2772$

## Mordell-Weil group structure

$\Z/{4}\Z$

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$\left(12, 6\right)$

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$\left(11, 0\right)$, $\left(12, 6\right)$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $48$ = $2^{4} \cdot 3$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $18432$ = $2^{11} \cdot 3^{2}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $\frac{3065617154}{9}$ = $2 \cdot 3^{-2} \cdot 1153^{3}$ $\text{End} (E)$ = $\Z$ (no Complex Multiplication) $\text{ST} (E)$ = $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() $r$ = $0$ magma: Regulator(E); sage: E.regulator() $\text{Reg}$ = $1$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] $\Omega$ ≈ $3.37150070963$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $4$  = $2\cdot2$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] $\#E_{\text{tor}}$ = $4$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Ш$_{\text{an}}$ = $1$ (exact)

## Modular invariants

### Modular form48.2.1.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$q + q^{3} - 2q^{5} + q^{9} - 4q^{11} - 2q^{13} - 2q^{15} + 2q^{17} + 4q^{19} + O(q^{20})$

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
8 : curve is not $\Gamma_0(N)$-optimal

### Special L-value attached to the curve

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

$L(E,1)$ ≈ $0.842875177406$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_3^{*}$ Additive 1 4 11 0
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X234b.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right)$ and has index 96.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $p$ Galois representation has maximal image $\GL(2,\F_p)$ for all primes $p$ except those listed.

prime Image of Galois representation
$2$ B

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type 2 3 add split - 1 - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 48.a consists of 6 curves linked by isogenies of degrees dividing 8.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $\Q(\sqrt{2})$ $\Z/2\Z \times \Z/4\Z$ 2.2.8.1-144.1-b7
$\Q(\sqrt{3})$ $\Z/8\Z$ 2.2.12.1-24.1-b7
$\Q(\sqrt{6})$ $\Z/8\Z$ 2.2.24.1-48.1-b6
4 $\Q(\sqrt{2}, \sqrt{3})$ $\Z/2\Z \times \Z/8\Z$ 4.4.2304.1-72.1-a7
4.4.13824.1 $\Z/16\Z$ 4.4.13824.1-24.1-e2

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.