# Properties

 Label 4290s1 Conductor $4290$ Discriminant $-2.602\times 10^{17}$ j-invariant $$-\frac{113180217375258301213009}{260161419375000000}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -1007741, -390570541])

gp: E = ellinit([1, 1, 1, -1007741, -390570541])

magma: E := EllipticCurve([1, 1, 1, -1007741, -390570541]);

$$y^2+xy+y=x^3+x^2-1007741x-390570541$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(1181, 7670\right)$$ $\hat{h}(P)$ ≈ $5.0977360456980663683578003414$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(1159, -580\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(1159, -580\right)$$, $$\left(1181, 7670\right)$$, $$\left(1181, -8852\right)$$, $$\left(138659, 51561920\right)$$, $$\left(138659, -51700580\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$4290$$ = $2 \cdot 3 \cdot 5 \cdot 11 \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-260161419375000000$ = $-1 \cdot 2^{6} \cdot 3^{7} \cdot 5^{10} \cdot 11^{4} \cdot 13$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{113180217375258301213009}{260161419375000000}$$ = $-1 \cdot 2^{-6} \cdot 3^{-7} \cdot 5^{-10} \cdot 11^{-4} \cdot 13^{-1} \cdot 2131^{3} \cdot 22699^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.2229114103670349530045365226\dots$ Stable Faltings height: $2.2229114103670349530045365226\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $5.0977360456980663683578003414\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.075330788672526051238583463260\dots$ 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: $48$  = $( 2 \cdot 3 )\cdot1\cdot2\cdot2^{2}\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ 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: $L'(E,1)$ ≈ $4.6081977212015957120952359913683847640$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 80640 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$3$ $1$ $I_{7}$ Non-split multiplicative 1 1 7 7
$5$ $2$ $I_{10}$ Non-split multiplicative 1 1 10 10
$11$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$13$ $1$ $I_{1}$ 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 2.3.0.1

## $p$-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 nonsplit nonsplit ordinary split nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss 2 1 5 1 2 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

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 4290s consists of 2 curves linked by isogenies of degree 2.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-39})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.2.62400.4 $$\Z/4\Z$$ Not in database $8$ 8.0.90079840281600.33 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.5922408960000.23 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.2.46297545707716875.10 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.