# Properties

 Label 4830bc2 Conductor 4830 Discriminant 5418996968417034240000 j-invariant $$\frac{9378698233516887309850369}{5418996968417034240000}$$ CM no Rank 0 Torsion Structure $$\Z/{2}\Z \times \Z/{6}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -4393456, -140558080]) # or

sage: E = EllipticCurve("4830bc2")

gp: E = ellinit([1, 0, 0, -4393456, -140558080]) \\ or

gp: E = ellinit("4830bc2")

magma: E := EllipticCurve([1, 0, 0, -4393456, -140558080]); // or

magma: E := EllipticCurve("4830bc2");

$$y^2 + x y = x^{3} - 4393456 x - 140558080$$

## Mordell-Weil group structure

$$\Z/{2}\Z \times \Z/{6}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-32, 16\right)$$, $$\left(11168, 1153616\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2080, 1040\right)$$, $$\left(-1432, 57416\right)$$, $$\left(-1432, -55984\right)$$, $$\left(-424, 40784\right)$$, $$\left(-424, -40360\right)$$, $$\left(-32, 16\right)$$, $$\left(3104, 125456\right)$$, $$\left(3104, -128560\right)$$, $$\left(11168, 1153616\right)$$, $$\left(11168, -1164784\right)$$

## 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: $$5418996968417034240000$$ = $$2^{18} \cdot 3^{12} \cdot 5^{4} \cdot 7^{6} \cdot 23^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{9378698233516887309850369}{5418996968417034240000}$$ = $$2^{-18} \cdot 3^{-12} \cdot 5^{-4} \cdot 7^{-6} \cdot 23^{-2} \cdot 43^{3} \cdot 193^{3} \cdot 25411^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.113958070835$$ 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: $$5184$$  = $$( 2 \cdot 3^{2} )\cdot( 2^{2} \cdot 3 )\cdot2\cdot( 2 \cdot 3 )\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$12$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form4830.2.a.be

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 331776 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$4.10249055007$$

## Local data

This elliptic curve is semistable.

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
$$3$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$5$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$7$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$23$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X8c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right)$ and has index 12.

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
$$2$$ Cs
$$3$$ B.1.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 23 split split nonsplit split nonsplit 4 7 0 1 0 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 4830bc consists of 8 curves linked by isogenies of degrees dividing 12.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{2}, \sqrt{-7})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{-2}, \sqrt{-23})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{7}, \sqrt{23})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.4722316875.1 $$\Z/6\Z \times \Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.