# Properties

 Label 950e3 Conductor $950$ Discriminant $-3.985\times 10^{13}$ j-invariant $$-\frac{69173457625}{2550136832}$$ CM no Rank $1$ Torsion structure trivial

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -2138, -306969])

gp: E = ellinit([1, 1, 1, -2138, -306969])

magma: E := EllipticCurve([1, 1, 1, -2138, -306969]);

$$y^2+xy+y=x^3+x^2-2138x-306969$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(105, 747\right)$$ $$\hat{h}(P)$$ ≈ $0.18876038702382457887589578183$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(81, 203\right)$$, $$\left(81, -285\right)$$, $$\left(105, 747\right)$$, $$\left(105, -853\right)$$, $$\left(169, 1963\right)$$, $$\left(169, -2133\right)$$, $$\left(1705, 69547\right)$$, $$\left(1705, -71253\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$950$$ = $$2 \cdot 5^{2} \cdot 19$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-39845888000000$$ = $$-1 \cdot 2^{27} \cdot 5^{6} \cdot 19$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{69173457625}{2550136832}$$ = $$-1 \cdot 2^{-27} \cdot 5^{3} \cdot 19^{-1} \cdot 821^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.2898885671742937682523415867\dots$$ Stable Faltings height: $$0.48516961095724358095196192009\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.18876038702382457887589578183\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.28183881244019952152146045064\dots$$ 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: $$54$$  = $$3^{3}\cdot2\cdot1$$ 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: $$1$$ (exact)

## Modular invariants

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);

$$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} - 2q^{9} - 6q^{11} - q^{12} - 5q^{13} + q^{14} + q^{16} - 3q^{17} - 2q^{18} + q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2592 $$\Gamma_0(N)$$-optimal: no 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);

$$L'(E,1)$$ ≈ $$2.8728001789855470210713058509642282711$$

## Local data

This elliptic curve is not semistable. There are 3 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$$(j)_{-}$$
$$2$$ $$27$$ $$I_{27}$$ Split multiplicative -1 1 27 27
$$5$$ $$2$$ $$I_0^{*}$$ Additive 1 2 6 0
$$19$$ $$1$$ $$I_{1}$$ 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(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 split ordinary add ordinary ordinary ordinary ordinary split ordinary ordinary ordinary ordinary ss ordinary ss 3 1 - 1 1 1 1 2 1 1 1 1 1,1 1 1,1 0 2 - 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 and 9.
Its isogeny class 950e consists of 3 curves linked by isogenies of degrees dividing 9.

## 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 $K$ $2$ $$\Q(\sqrt{-15})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.152.1 $$\Z/2\Z$$ Not in database $6$ 6.0.3511808.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.146611125.2 $$\Z/3\Z$$ Not in database $6$ 6.0.320638530375.8 $$\Z/9\Z$$ Not in database $6$ 6.0.77976000.2 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.2.298227647455089932395008000000000.1 $$\Z/6\Z$$ Not in database $18$ 18.0.3119566527280004772981175394010624000000000.1 $$\Z/18\Z$$ 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.