# Properties

 Label 31200.a2 Conductor $31200$ Discriminant $1.028\times 10^{13}$ j-invariant $$\frac{33324076232}{1285245}$$ CM no Rank $1$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -13408, 581812])

gp: E = ellinit([0, -1, 0, -13408, 581812])

magma: E := EllipticCurve([0, -1, 0, -13408, 581812]);

$$y^2=x^3-x^2-13408x+581812$$

## Mordell-Weil group structure

$\Z\times \Z/{4}\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(52, 150\right)$$ (52, 150) $\hat{h}(P)$ ≈ $1.3533013658469342342091192572$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(12, 650\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-92,\pm 1014)$$, $$(12,\pm 650)$$, $$(52,\pm 150)$$, $$\left(77, 0\right)$$, $$(221,\pm 2892)$$, $$(337,\pm 5850)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$31200$$ = $2^{5} \cdot 3 \cdot 5^{2} \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $10281960000000$ = $2^{9} \cdot 3^{2} \cdot 5^{7} \cdot 13^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{33324076232}{1285245}$$ = $2^{3} \cdot 3^{-2} \cdot 5^{-1} \cdot 13^{-4} \cdot 1609^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.2644656268961680252433980885\dots$ Stable Faltings height: $-0.060113714740841144119905669206\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.3533013658469342342091192572\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.71725366599731205108378246481\dots$ 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: $64$  = $2\cdot2\cdot2^{2}\cdot2^{2}$ 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: $L'(E,1)$ ≈ $3.8826414634115326776309976835$

## Modular invariants

Modular form 31200.2.a.a

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 73728 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $I_0^{*}$ Additive -1 5 9 0
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$5$ $4$ $I_{1}^{*}$ Additive 1 2 7 1
$13$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

## 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
$2$ 2B 4.12.0.7

## $p$-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 nonsplit add ord ord split ord ord ss ord ss ord ord ord ord - 1 - 1 1 2 1 1 1,1 1 1,1 1 1 1 3 - 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 and 4.
Its isogeny class 31200.a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{10})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.0.10816000.3 $$\Z/8\Z$$ Not in database $8$ 8.0.21233664000000.33 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.0.1871773696000000.53 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.8.6064546775040000.1 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ 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.