# Properties

 Label 379050.s1 Conductor $379050$ Discriminant $-4.502\times 10^{37}$ j-invariant $$-\frac{109264302241400105173004689}{169637740417250683593750}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -640031191275, 378214736430328875])

gp: E = ellinit([1, 1, 0, -640031191275, 378214736430328875])

magma: E := EllipticCurve([1, 1, 0, -640031191275, 378214736430328875]);

$$y^2+xy=x^3+x^2-640031191275x+378214736430328875$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-\frac{290656082083318389747344417178558705}{695830692062434107901983436036}, \frac{439370741809256928912689542236953278495209546461590785}{580437377695808013131988505258853443861825784}\right)$$ $\hat{h}(P)$ ≈ $72.858630714334881462934845051$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$379050$$ = $2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-45016457164205791073668711853027343750$ = $-1 \cdot 2 \cdot 3^{16} \cdot 5^{17} \cdot 7^{9} \cdot 19^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{109264302241400105173004689}{169637740417250683593750}$$ = $-1 \cdot 2^{-1} \cdot 3^{-16} \cdot 5^{-11} \cdot 7^{-9} \cdot 19 \cdot 67^{3} \cdot 229^{3} \cdot 11677^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $5.9164132644002006778947760808\dots$ Stable Faltings height: $3.1487349887388568505883781263\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $72.858630714334881462934845051\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.0057395838581013694382939905879\dots$ 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: $12$  = $1\cdot2\cdot2\cdot1\cdot3$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ 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: $L'(E,1)$ ≈ $5.0181386492563815998566281493412582417$

## Modular invariants

Modular form 379050.2.a.s

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);

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

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 15081707520 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$3$ $2$ $I_{16}$ Non-split multiplicative 1 1 16 16
$5$ $2$ $I_{11}^{*}$ Additive 1 2 17 11
$7$ $1$ $I_{9}$ Non-split multiplicative 1 1 9 9
$19$ $3$ $IV^{*}$ Additive 1 2 8 0

## 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 $\ell$.

## $p$-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.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has no rational isogenies. Its isogeny class 379050.s consists of this curve only.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.101080.1 $$\Z/2\Z$$ Not in database $6$ 6.0.2860806592000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.5771493546750000.6 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.