# Properties

 Label 4830.bk3 Conductor $4830$ Discriminant $-2.293\times 10^{13}$ j-invariant $$-\frac{12232183057921}{22933241856000}$$ CM no Rank $1$ Torsion structure $$\Z/{6}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -480, 230400])

gp: E = ellinit([1, 0, 0, -480, 230400])

magma: E := EllipticCurve([1, 0, 0, -480, 230400]);

$$y^2+xy=x^3-480x+230400$$

## Mordell-Weil group structure

$\Z\times \Z/{6}\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(80, 800\right)$$ $\hat{h}(P)$ ≈ $0.66728020879171298220115793737$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(120, 1320\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-64, 32\right)$$, $$\left(-60, 240\right)$$, $$\left(-60, -180\right)$$, $$\left(-18, 492\right)$$, $$\left(-18, -474\right)$$, $$\left(0, 480\right)$$, $$\left(0, -480\right)$$, $$\left(36, 492\right)$$, $$\left(36, -528\right)$$, $$\left(80, 800\right)$$, $$\left(80, -880\right)$$, $$\left(120, 1320\right)$$, $$\left(120, -1440\right)$$, $$\left(192, 2592\right)$$, $$\left(192, -2784\right)$$, $$\left(350, 6380\right)$$, $$\left(350, -6730\right)$$, $$\left(720, 18960\right)$$, $$\left(720, -19680\right)$$, $$\left(2880, 153120\right)$$, $$\left(2880, -156000\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$4830$$ = $2 \cdot 3 \cdot 5 \cdot 7 \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-22933241856000$ = $-1 \cdot 2^{18} \cdot 3^{3} \cdot 5^{3} \cdot 7^{2} \cdot 23^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{12232183057921}{22933241856000}$$ = $-1 \cdot 2^{-18} \cdot 3^{-3} \cdot 5^{-3} \cdot 7^{-2} \cdot 23^{-2} \cdot 23041^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.2426055356389429050899460883\dots$ Stable Faltings height: $1.2426055356389429050899460883\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.66728020879171298220115793737\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.54428260800525884054107736003\dots$ 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: $648$  = $( 2 \cdot 3^{2} )\cdot3\cdot3\cdot2\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $6$ 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: $L'(E,1)$ ≈ $6.5374022216060494371748124720042261909$

## 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);

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - 6 q^{11} + q^{12} - 4 q^{13} + q^{14} + q^{15} + q^{16} - 6 q^{17} + q^{18} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 20736 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is 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$(j)_{-}$
$2$ $18$ $I_{18}$ Split multiplicative -1 1 18 18
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$5$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$23$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## 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 2.3.0.1
$3$ 3B.1.1 3.8.0.1

## $p$-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 split split split split ordinary ordinary ordinary ordinary nonsplit ss ordinary ordinary ordinary ordinary ss 4 2 2 2 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

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 4830.bk consists of 4 curves linked by isogenies of degrees dividing 6.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-15})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $4$ 4.2.507840.4 $$\Z/12\Z$$ Not in database $6$ 6.0.18141252507.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $8$ 8.0.58027829760000.7 $$\Z/2\Z \times \Z/12\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/12\Z$$ Not in database $9$ 9.3.150640708962071671875.1 $$\Z/18\Z$$ Not in database $12$ Deg 12 $$\Z/6\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $18$ 18.0.76587603288510084260296309282207122802734375.1 $$\Z/2\Z \times \Z/18\Z$$ 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.