# Properties

 Label 46800fb2 Conductor $46800$ Discriminant $8.277\times 10^{20}$ j-invariant $$\frac{38686490446661}{141927552}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -12682875, 17329806250])

gp: E = ellinit([0, 0, 0, -12682875, 17329806250])

magma: E := EllipticCurve([0, 0, 0, -12682875, 17329806250]);

$$y^2=x^3-12682875x+17329806250$$

## Mordell-Weil group structure

$\Z^2 \times \Z/{2}\Z$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(525, 104000\right)$$ $$\left(2189, 7488\right)$$ $\hat{h}(P)$ ≈ $1.5945928646617590328535614477$ $2.0759388709559482786222507563$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(2150, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-1009,\pm 170586)$$, $$(525,\pm 104000)$$, $$(650,\pm 96750)$$, $$(1421,\pm 46656)$$, $$(1775,\pm 20250)$$, $$\left(2150, 0\right)$$, $$(2189,\pm 7488)$$, $$(2199,\pm 8582)$$, $$(8391,\pm 708314)$$, $$(32525,\pm 5832000)$$, $$(2578061,\pm 4139418816)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$46800$$ = $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $827721483264000000000$ = $2^{19} \cdot 3^{14} \cdot 5^{9} \cdot 13^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{38686490446661}{141927552}$$ = $2^{-7} \cdot 3^{-8} \cdot 13^{-2} \cdot 31^{3} \cdot 1091^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.8741999008712539177255448042\dots$ Stable Faltings height: $0.42466814165167848166012056436\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $3.2000838575208064784955424663\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.15935684818192926122363429234\dots$ 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: $64$  = $2^{2}\cdot2^{2}\cdot2\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (rounded) 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: $L^{(2)}(E,1)/2!$ ≈ $8.1592844392381713101354805300007708746$

## Modular invariants

Modular form 46800.2.a.e

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);

$$q - 4q^{7} - 6q^{11} - q^{13} - 4q^{17} - 2q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3440640 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$2$ $4$ $I_{11}^{*}$ Additive -1 4 19 7
$3$ $4$ $I_8^{*}$ Additive -1 2 14 8
$5$ $2$ $III^{*}$ Additive -1 2 9 0
$13$ $2$ $I_{2}$ Non-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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1

## $p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

## 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 add add ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary - - - 2 2 2 2 4 2 2 2 2 2 2,2 2 - - - 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 46800fb consists of 2 curves linked by isogenies of degree 2.

## 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 $K$ $2$ $$\Q(\sqrt{10})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.4.1521000.2 $$\Z/4\Z$$ Not in database $8$ 8.8.148060224000000.12 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.