# Properties

 Label 4830ba2 Conductor $4830$ Discriminant $4.115\times 10^{12}$ j-invariant $$\frac{2191574502231419089}{4115217960000}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -27061, 1708385])

gp: E = ellinit([1, 0, 0, -27061, 1708385])

magma: E := EllipticCurve([1, 0, 0, -27061, 1708385]);

## Simplified equation

 $$y^2+xy=x^3-27061x+1708385$$ y^2+xy=x^3-27061x+1708385 (homogenize, simplify) $$y^2z+xyz=x^3-27061xz^2+1708385z^3$$ y^2z+xyz=x^3-27061xz^2+1708385z^3 (dehomogenize, simplify) $$y^2=x^3-35071083x+79811623782$$ y^2=x^3-35071083x+79811623782 (homogenize, minimize)

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(86, 95\right)$$ (86, 95) $\hat{h}(P)$ ≈ $0.63481666467750586994754056581$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-190, 95\right)$$, $$\left(98, -49\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-190, 95\right)$$, $$\left(-52, 1751\right)$$, $$\left(-52, -1699\right)$$, $$\left(86, 95\right)$$, $$\left(86, -181\right)$$, $$\left(98, -49\right)$$, $$\left(104, 95\right)$$, $$\left(104, -199\right)$$, $$\left(134, 635\right)$$, $$\left(134, -769\right)$$, $$\left(148, 901\right)$$, $$\left(148, -1049\right)$$, $$\left(398, 7151\right)$$, $$\left(398, -7549\right)$$

## 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: $4115217960000$ = $2^{6} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 23^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2191574502231419089}{4115217960000}$$ = $2^{-6} \cdot 3^{-4} \cdot 5^{-4} \cdot 7^{-4} \cdot 23^{-2} \cdot 67^{3} \cdot 19387^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.3107078129722741868440924453\dots$ Stable Faltings height: $1.3107078129722741868440924453\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.63481666467750586994754056581\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.78114409035795583736095163298\dots$ 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: $192$  = $( 2 \cdot 3 )\cdot2^{2}\cdot2\cdot2\cdot2$ 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: $L'(E,1)$ ≈ $5.9505994328829815643018358023$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 18432 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$2$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$3$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$5$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$7$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$23$ $2$ $I_{2}$ 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
$2$ 2Cs 8.24.0.1

## $p$-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 split nonsplit nonsplit ss ord ord ss nonsplit ord ord ord ord ss ord 4 2 1 1 1,1 1 3 1,1 1 3 1 1 1 1,1 1 0 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.
Its isogeny class 4830ba 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{-23})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{23})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.18339659776.1 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.4.112326082560000.21 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.74390473561516875.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ 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.