# Properties

 Label 379050.c1 Conductor $379050$ Discriminant $-3.353\times 10^{27}$ j-invariant $$-\frac{1569510182075597}{36491419872936}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -109252325, 2820401947125])

gp: E = ellinit([1, 1, 0, -109252325, 2820401947125])

magma: E := EllipticCurve([1, 1, 0, -109252325, 2820401947125]);

$$y^2+xy=x^3+x^2-109252325x+2820401947125$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{1695215}{49}, \frac{2175567185}{343}\right)$$ $\hat{h}(P)$ ≈ $7.5661340924552309415987398356$

## 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: $-3353068353248402688703125000$ = $-1 \cdot 2^{3} \cdot 3^{6} \cdot 5^{9} \cdot 7 \cdot 19^{13}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1569510182075597}{36491419872936}$$ = $-1 \cdot 2^{-3} \cdot 3^{-6} \cdot 7^{-1} \cdot 19^{-7} \cdot 251^{3} \cdot 463^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.9623717554837223533636746072\dots$ Stable Faltings height: $1.2830738315749268424085913913\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $7.5661340924552309415987398356\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.037452146445554875503708072579\dots$ 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: $8$  = $1\cdot2\cdot2\cdot1\cdot2$ 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: $L'(E,1)$ ≈ $2.2669436964587099303738013088349155384$

## Modular invariants

Modular form 379050.2.a.c

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 275788800 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$3$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$5$ $2$ $III^{*}$ Additive -1 2 9 0
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$19$ $2$ $I_7^{*}$ Additive -1 2 13 7

## 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 $\GL(2,\Z_\ell)$ 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.c 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 $K$ $3$ 3.1.5320.1 $$\Z/2\Z$$ Not in database $6$ 6.0.150568768000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ 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.