# Properties

 Label 303450.q1 Conductor $303450$ Discriminant $-182070000$ j-invariant $$-\frac{2088025}{1008}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -150, 900])

gp: E = ellinit([1, 1, 0, -150, 900])

magma: E := EllipticCurve([1, 1, 0, -150, 900]);

$$y^2+xy=x^3+x^2-150x+900$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(0, 30\right)$$ $$\left(20, 70\right)$$ $$\hat{h}(P)$$ ≈ $0.35940742612019859978318421993$ $1.3962918180239015317109798834$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-15, 15\right)$$, $$\left(-15, 0\right)$$, $$\left(0, 30\right)$$, $$\left(0, -30\right)$$, $$\left(5, 15\right)$$, $$\left(5, -20\right)$$, $$\left(9, 15\right)$$, $$\left(9, -24\right)$$, $$\left(16, 46\right)$$, $$\left(16, -62\right)$$, $$\left(20, 70\right)$$, $$\left(20, -90\right)$$, $$\left(84, 726\right)$$, $$\left(84, -810\right)$$, $$\left(1085, 35215\right)$$, $$\left(1085, -36300\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$303450$$ = $$2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-182070000$$ = $$-1 \cdot 2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7 \cdot 17^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{2088025}{1008}$$ = $$-1 \cdot 2^{-4} \cdot 3^{-2} \cdot 5^{2} \cdot 7^{-1} \cdot 17^{4}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.29059175075057693549505448040\dots$$ Stable Faltings height: $$-0.71808977740349253608012106699\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.48284205305043571176472758359\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$1.6789913570338399266173781795\dots$$ 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: $$12$$  = $$2\cdot2\cdot3\cdot1\cdot1$$ 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 303450.2.a.q

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);

$$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9} + 3q^{11} - q^{12} + q^{14} + q^{16} - q^{18} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 124416 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$9.7282516066098766173615815611246789012$$

## Local data

This elliptic curve is not semistable. There are 5 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$5$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$17$$ $$1$$ $$II$$ Additive 1 2 2 0

## 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 $$p$$ .

## $p$-adic data

### $p$-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.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has no rational isogenies. Its isogeny class 303450.q consists of this curve only.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.50575.2 $$\Z/2\Z$$ Not in database $6$ 6.0.17904814375.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.