# Properties

 Label 4620.n1 Conductor $4620$ Discriminant $-2.376\times 10^{12}$ j-invariant $$-\frac{4890195460096}{9282994875}$$ CM no Rank $1$ Torsion structure $$\Z/{3}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3+x^2-2245x+83975$$ y^2=x^3+x^2-2245x+83975 (homogenize, simplify) $$y^2z=x^3+x^2z-2245xz^2+83975z^3$$ y^2z=x^3+x^2z-2245xz^2+83975z^3 (dehomogenize, simplify) $$y^2=x^3-181872x+61763364$$ y^2=x^3-181872x+61763364 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -2245, 83975])

gp: E = ellinit([0, 1, 0, -2245, 83975])

magma: E := EllipticCurve([0, 1, 0, -2245, 83975]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z \oplus \Z/{3}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(50, 315\right)$$ (50, 315) $\hat{h}(P)$ ≈ $0.23832444413787705089534664050$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(5, 270\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-55,\pm 210)$$, $$(-22,\pm 351)$$, $$(5,\pm 270)$$, $$(25,\pm 210)$$, $$(29,\pm 210)$$, $$(50,\pm 315)$$, $$(113,\pm 1134)$$, $$(365,\pm 6930)$$, $$(545,\pm 12690)$$, $$(10970,\pm 1149015)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$4620$$ = $2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-2376446688000$ = $-1 \cdot 2^{8} \cdot 3^{9} \cdot 5^{3} \cdot 7^{3} \cdot 11$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{4890195460096}{9282994875}$$ = $-1 \cdot 2^{16} \cdot 3^{-9} \cdot 5^{-3} \cdot 7^{-3} \cdot 11^{-1} \cdot 421^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.0645077817841732981821979203\dots$ Stable Faltings height: $0.60240966141087642523737650600\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.23832444413787705089534664050\dots$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.72872311336521791304126503295\dots$ 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: $243$  = $3\cdot3^{2}\cdot3\cdot3\cdot1$ 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) 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: $L'(E,1)$ ≈ $4.6891583349260954900707290932$

## 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);

$$q + q^{3} + q^{5} + q^{7} + q^{9} - q^{11} - 4 q^{13} + q^{15} + 3 q^{17} - 7 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 7776 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## 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$(j)_{-}$
$2$ $3$ $IV^{*}$ Additive -1 2 8 0
$3$ $9$ $I_{9}$ Split multiplicative -1 1 9 9
$5$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$11$ $1$ $I_{1}$ 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
$3$ 3B.1.1 3.8.0.1
sage: gens = [[4, 3, 9, 7], [2305, 6, 2304, 7], [661, 6, 1983, 19], [1387, 6, 1851, 19], [211, 6, 633, 19], [1, 6, 0, 1], [3, 4, 8, 11], [1, 0, 6, 1], [1926, 391, 1543, 1944]]

sage: GL(2,Integers(2310)).subgroup(gens)

magma: Gens := [[4, 3, 9, 7], [2305, 6, 2304, 7], [661, 6, 1983, 19], [1387, 6, 1851, 19], [211, 6, 633, 19], [1, 6, 0, 1], [3, 4, 8, 11], [1, 0, 6, 1], [1926, 391, 1543, 1944]];

magma: sub<GL(2,Integers(2310))|Gens>;

The image of the adelic Galois representation has level $2310$, index $16$, genus $0$, and generators

$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 2305 & 6 \\ 2304 & 7 \end{array}\right),\left(\begin{array}{rr} 661 & 6 \\ 1983 & 19 \end{array}\right),\left(\begin{array}{rr} 1387 & 6 \\ 1851 & 19 \end{array}\right),\left(\begin{array}{rr} 211 & 6 \\ 633 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 1926 & 391 \\ 1543 & 1944 \end{array}\right)$

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $p$-adic regulator data only exists for primes $p\ge 5$ 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 add split split split nonsplit ord ord ord ord ord ord ord ord ord ord - 6 2 2 1 1 1 1 3 1 3 1 1 3 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 4620.n 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 $K$ $3$ 3.1.4620.1 $$\Z/6\Z$$ Not in database $6$ 6.0.24652782000.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $6$ 6.0.6324912.1 $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $9$ 9.3.288178803000000.7 $$\Z/9\Z$$ Not in database $12$ deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.56280390065562432115862208000000.1 $$\Z/3\Z \oplus \Z/6\Z$$ 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.