# Properties

 Label 100800.d2 Conductor 100800 Discriminant -180592312320000 j-invariant $$\frac{2595575}{1512}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 14100, -52400]) # or

sage: E = EllipticCurve("100800pd1")

gp: E = ellinit([0, 0, 0, 14100, -52400]) \\ or

gp: E = ellinit("100800pd1")

magma: E := EllipticCurve([0, 0, 0, 14100, -52400]); // or

magma: E := EllipticCurve("100800pd1");

$$y^2 = x^{3} + 14100 x - 52400$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(5, 135\right)$$ $$\left(26, 576\right)$$ $$\hat{h}(P)$$ ≈ 1.7881676843821084 1.1760114055199553

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(5,\pm 135)$$, $$(26,\pm 576)$$, $$(80,\pm 1260)$$, $$(140,\pm 2160)$$, $$(144,\pm 2228)$$, $$(410,\pm 8640)$$, $$(1050,\pm 34240)$$, $$(3029,\pm 166833)$$, $$(1203125,\pm 1319672385)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$100800$$ = $$2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-180592312320000$$ = $$-1 \cdot 2^{21} \cdot 3^{9} \cdot 5^{4} \cdot 7$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2595575}{1512}$$ = $$2^{-3} \cdot 3^{-3} \cdot 5^{2} \cdot 7^{-1} \cdot 47^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.663084827052$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.336495494267$$ 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: $$48$$  = $$2^{2}\cdot2^{2}\cdot3\cdot1$$ 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: $$1$$ (rounded)

## Modular invariants

#### Modular form 100800.2.a.d

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);

$$q - q^{7} - 6q^{11} + q^{13} - 3q^{17} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 331776 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

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);

$$L^{(2)}(E,1)/2!$$ ≈ $$10.7100027177$$

## Local data

This elliptic curve is not semistable.

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 6 21 3
$$3$$ $$4$$ $$I_3^{*}$$ Additive -1 2 9 3
$$5$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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

prime Image of Galois representation
$$3$$ B

## $p$-adic data

### $p$-adic regulators

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

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ 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 add add nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss - - - 4 2 2 2 2 2 2 4 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 $$d=$$ 3.
Its isogeny class 100800.d consists of 2 curves linked by isogenies of degree 3.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{6})$$ $$\Z/3\Z$$ Not in database
3 3.1.4200.1 $$\Z/2\Z$$ Not in database
6 6.2.423360000.1 $$\Z/6\Z$$ Not in database
6.0.2963520000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database
6.0.6914880000.2 $$\Z/3\Z$$ Not in database

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.