# Properties

 Label 1248.b3 Conductor $1248$ Discriminant $7884864$ j-invariant $$\frac{22235451328}{123201}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-x^2-234x-1296$$ y^2=x^3-x^2-234x-1296 (homogenize, simplify) $$y^2z=x^3-x^2z-234xz^2-1296z^3$$ y^2z=x^3-x^2z-234xz^2-1296z^3 (dehomogenize, simplify) $$y^2=x^3-18981x-1001700$$ y^2=x^3-18981x-1001700 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, -234, -1296])

gp: E = ellinit([0, -1, 0, -234, -1296])

magma: E := EllipticCurve([0, -1, 0, -234, -1296]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(234, 3564\right)$$ (234, 3564) $\hat{h}(P)$ ≈ $3.4964988975274232305095089727$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-8, 0\right)$$, $$\left(18, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-9, 0\right)$$, $$\left(-8, 0\right)$$, $$\left(18, 0\right)$$, $$(234,\pm 3564)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1248$$ = $2^{5} \cdot 3 \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $7884864$ = $2^{6} \cdot 3^{6} \cdot 13^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{22235451328}{123201}$$ = $2^{6} \cdot 3^{-6} \cdot 13^{-2} \cdot 19^{3} \cdot 37^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.16614275206940871568696688812\dots$ Stable Faltings height: $-0.18043083821056393902164917261\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $3.4964988975274232305095089727\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $1.2206352993932699825054064318\dots$ 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: $8$  = $2\cdot2\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $4$ 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: $L'(E,1)$ ≈ $2.1339749893058123379666515086$

## 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);

$$q - q^{3} - 2 q^{5} + q^{9} + 4 q^{11} + q^{13} + 2 q^{15} - 6 q^{17} - 8 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 384 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$2$ $2$ $III$ Additive -1 5 6 0
$3$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$13$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

## 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 4.12.0.1

## $p$-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 nonsplit ord ss ord split ord ord ord ord ord ord ord ord ss - 1 7 1,1 1 2 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 non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 1248.b 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-26})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{26})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\zeta_{12})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.151613669376.9 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.2915295018614784.19 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.4093569073152.30 $$\Z/2\Z \oplus \Z/6\Z$$ 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 $$\Z/2\Z \oplus \Z/12\Z$$ 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.