# Properties

 Label 75150b1 Conductor 75150 Discriminant -5030060040000000 j-invariant $$-\frac{77325109990227}{11923105280}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -66567, -7422659]); // or

magma: E := EllipticCurve("75150b1");

sage: E = EllipticCurve([1, -1, 0, -66567, -7422659]) # or

sage: E = EllipticCurve("75150b1")

gp: E = ellinit([1, -1, 0, -66567, -7422659]) \\ or

gp: E = ellinit("75150b1")

$$y^2 + x y = x^{3} - x^{2} - 66567 x - 7422659$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(549, 10738\right)$$ $$\hat{h}(P)$$ ≈ 4.54129749018

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(549, 10738\right)$$

Note: only one of each pair $\pm P$ is listed.

## 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: $$-5030060040000000$$ = $$-1 \cdot 2^{9} \cdot 3^{3} \cdot 5^{7} \cdot 167^{3}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{77325109990227}{11923105280}$$ = $$-1 \cdot 2^{-9} \cdot 3^{3} \cdot 5^{-1} \cdot 11^{3} \cdot 167^{-3} \cdot 1291^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$4.54129749018$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.147359999384$$ 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: $$8$$  = $$1\cdot2\cdot2^{2}\cdot1$$ 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: $$1$$ (exact)

## Modular invariants

#### Modular form 75150.2.a.t

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)

$$q - q^{2} + q^{4} + q^{7} - q^{8} + 4q^{13} - q^{14} + q^{16} + 5q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 414720 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$5.35364476284$$

## 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{9}$$ Non-split multiplicative 1 1 9 9
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$5$$ $$4$$ $$I_1^{*}$$ Additive 1 2 7 1
$$167$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3

## 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 $$p$$ 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 47 167 nonsplit add add ordinary ss ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit 4 - - 1 1,1 1 1,1 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 $$d=$$ 3.
Its isogeny class 75150b 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 $$\Q(\sqrt{5})$$ $$\Z/3\Z$$ Not in database
3 3.1.20040.1 $$\Z/2\Z$$ Not in database
6 6.2.2008008000.1 $$\Z/6\Z$$ Not in database
6.0.6834375.1 $$\Z/3\Z$$ Not in database
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.