Properties

 Label 4560g4 Conductor $4560$ Discriminant $615600000000$ j-invariant $$\frac{1280615525284}{601171875}$$ CM no Rank $1$ Torsion structure $$\Z/{4}\Z$$

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -2280, -18972])

gp: E = ellinit([0, 1, 0, -2280, -18972])

magma: E := EllipticCurve([0, 1, 0, -2280, -18972]);

Simplified equation

 $$y^2=x^3+x^2-2280x-18972$$ y^2=x^3+x^2-2280x-18972 (homogenize, simplify) $$y^2z=x^3+x^2z-2280xz^2-18972z^3$$ y^2z=x^3+x^2z-2280xz^2-18972z^3 (dehomogenize, simplify) $$y^2=x^3-184707x-13276494$$ y^2=x^3-184707x-13276494 (homogenize, minimize)

Mordell-Weil group structure

$$\Z \oplus \Z/{4}\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-9, 30\right)$$ (-9, 30) $\hat{h}(P)$ ≈ $1.5905794107966419817031431735$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-24, 150\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-24,\pm 150)$$, $$(-9,\pm 30)$$, $$\left(51, 0\right)$$, $$(96,\pm 810)$$, $$(176,\pm 2250)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$4560$$ = $2^{4} \cdot 3 \cdot 5 \cdot 19$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $615600000000$ = $2^{10} \cdot 3^{4} \cdot 5^{8} \cdot 19$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1280615525284}{601171875}$$ = $2^{2} \cdot 3^{-4} \cdot 5^{-8} \cdot 19^{-1} \cdot 6841^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.95645126778008812672760591502\dots$ Stable Faltings height: $0.37882861731346703554657914714\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.5905794107966419817031431735\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.72311931828675310462836559021\dots$ 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: $64$  = $2\cdot2^{2}\cdot2^{3}\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $4$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) 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); Special value: $L'(E,1)$ ≈ $4.6007147968648526828129557655$

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);

$$q + q^{3} + q^{5} + q^{9} - 2 q^{13} + q^{15} - 6 q^{17} - q^{19} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4096 $\Gamma_0(N)$-optimal: no Manin constant: 1

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$(j)_{-}$
$2$ $2$ $I_{2}^{*}$ Additive 1 4 10 0
$3$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$5$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$19$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.12.0.7

$p$-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 add split split ss ss ord ord nonsplit ord ord ss ord ord ord ord - 2 2 1,3 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 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 $d=$ 2 and 4.
Its isogeny class 4560g consists of 4 curves linked by isogenies of degrees dividing 4.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{19})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ 4.0.10944.1 $$\Z/8\Z$$ Not in database $8$ 8.0.192699928576.2 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.8.249739107434496.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.691798081536.11 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ 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.