# Properties

 Label 75150z1 Conductor 75150 Discriminant -72144000000000 j-invariant $$\frac{190804533093}{171008000}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, 8995, 240997]); // or

magma: E := EllipticCurve("75150z1");

sage: E = EllipticCurve([1, -1, 1, 8995, 240997]) # or

sage: E = EllipticCurve("75150z1")

gp: E = ellinit([1, -1, 1, 8995, 240997]) \\ or

gp: E = ellinit("75150z1")

$$y^2 + x y + y = x^{3} - x^{2} + 8995 x + 240997$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-11, -370\right)$$ $$\left(-17, 296\right)$$ $$\hat{h}(P)$$ ≈ 0.676905990585 1.87831761953

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-25, 16\right)$$, $$\left(-25, 8\right)$$, $$\left(-17, 296\right)$$, $$\left(-17, -280\right)$$, $$\left(-11, 380\right)$$, $$\left(-11, -370\right)$$, $$\left(29, 710\right)$$, $$\left(29, -740\right)$$, $$\left(79, 1160\right)$$, $$\left(79, -1240\right)$$, $$\left(179, 2660\right)$$, $$\left(179, -2840\right)$$, $$\left(239, 3880\right)$$, $$\left(239, -4120\right)$$, $$\left(739, 19880\right)$$, $$\left(739, -20620\right)$$, $$\left(1303, 46496\right)$$, $$\left(1303, -47800\right)$$, $$\left(1375, 50408\right)$$, $$\left(1375, -51784\right)$$, $$\left(12239, 1347880\right)$$, $$\left(12239, -1360120\right)$$, $$\left(1859183, 2534100520\right)$$, $$\left(1859183, -2535959704\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$75150$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 167$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-72144000000000$$ = $$-1 \cdot 2^{13} \cdot 3^{3} \cdot 5^{9} \cdot 167$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{190804533093}{171008000}$$ = $$2^{-13} \cdot 3^{3} \cdot 5^{-3} \cdot 19^{3} \cdot 101^{3} \cdot 167^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.35766408436$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.40093665344$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$104$$  = $$13\cdot2\cdot2^{2}\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 75150.2.a.bf

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)

$$q + q^{2} + q^{4} - q^{7} + q^{8} - 4q^{13} - q^{14} + q^{16} - 8q^{17} - 7q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 239616 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

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])

$$L^{(2)}(E,1)/2!$$ ≈ $$14.9136666681$$

## 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$$(j)_{-}$$
$$2$$ $$13$$ $$I_{13}$$ Split multiplicative -1 1 13 13
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$5$$ $$4$$ $$I_3^{*}$$ Additive 1 2 9 3
$$167$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

## Galois representations

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

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 $$p$$ .

## $p$-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 47 167 split add add ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit 4 - - 2 2,2 2 2 2 2 2 2 2 2 2 2 2 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 no rational isogenies. Its isogeny class 75150z consists of this curve only.

## 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
3 3.1.20040.1 $$\Z/2\Z$$ Not in database
6 6.0.8048096064000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.