Properties

 Label 3150br2 Conductor $3150$ Discriminant $-240045120000$ j-invariant $$-\frac{7620530425}{526848}$$ CM no Rank $1$ Torsion structure $$\Z/{3}\Z$$

Related objects

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -3155, 72947])

gp: E = ellinit([1, -1, 1, -3155, 72947])

magma: E := EllipticCurve([1, -1, 1, -3155, 72947]);

$$y^2+xy+y=x^3-x^2-3155x+72947$$

Mordell-Weil group structure

$$\Z\times \Z/{3}\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(39, 70\right)$$ $$\hat{h}(P)$$ ≈ $0.13815541856997857598492782581$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(19, 130\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-65, 46\right)$$, $$\left(-65, 18\right)$$, $$\left(-51, 340\right)$$, $$\left(-51, -290\right)$$, $$\left(-21, 370\right)$$, $$\left(-21, -350\right)$$, $$\left(3, 250\right)$$, $$\left(3, -254\right)$$, $$\left(19, 130\right)$$, $$\left(19, -150\right)$$, $$\left(33, 46\right)$$, $$\left(33, -80\right)$$, $$\left(39, 70\right)$$, $$\left(39, -110\right)$$, $$\left(49, 160\right)$$, $$\left(49, -210\right)$$, $$\left(75, 466\right)$$, $$\left(75, -542\right)$$, $$\left(159, 1810\right)$$, $$\left(159, -1970\right)$$, $$\left(579, 13570\right)$$, $$\left(579, -14150\right)$$, $$\left(669, 16900\right)$$, $$\left(669, -17570\right)$$, $$\left(118123, 40538530\right)$$, $$\left(118123, -40656654\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$3150$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-240045120000$$ = $$-1 \cdot 2^{9} \cdot 3^{7} \cdot 5^{4} \cdot 7^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{7620530425}{526848}$$ = $$-1 \cdot 2^{-9} \cdot 3^{-1} \cdot 5^{2} \cdot 7^{-3} \cdot 673^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.93462356429511338019336642610\dots$$ Stable Faltings height: $$-0.15116188418364159037117597010\dots$$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.13815541856997857598492782581\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.97235150404184267083718384785\dots$$ 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: $$324$$  = $$3^{2}\cdot2^{2}\cdot3\cdot3$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$3$$ 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 5184 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$4.8360826453697636151715136329857148351$$

Local data

This elliptic curve is not semistable. There are 4 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$$(j)_{-}$$
$$2$$ $$9$$ $$I_{9}$$ Split multiplicative -1 1 9 9
$$3$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1
$$5$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$7$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3

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.1.1

$p$-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 add add split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss 2 - - 2 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

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 3150br consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.4200.1 $$\Z/6\Z$$ Not in database $6$ 6.0.2963520000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.0.110716875.2 $$\Z/3\Z \times \Z/3\Z$$ Not in database $9$ 9.3.26377102494421875.2 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.41857146662060206427328000000000000.3 $$\Z/3\Z \times \Z/6\Z$$ 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.