# Properties

 Label 379050.hv1 Conductor $379050$ Discriminant $-1.549\times 10^{16}$ j-invariant $$-\frac{7620530425}{526848}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -126538, 18278231])

gp: E = ellinit([1, 1, 1, -126538, 18278231])

magma: E := EllipticCurve([1, 1, 1, -126538, 18278231]);

$$y^2+xy+y=x^3+x^2-126538x+18278231$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-59, 5083\right)$$ $$\hat{h}(P)$$ ≈ $0.67144180902244104032824765632$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-59, 5083\right)$$, $$\left(-59, -5025\right)$$, $$\left(249, 1387\right)$$, $$\left(249, -1637\right)$$

## 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: $$-15491267695680000$$ = $$-1 \cdot 2^{9} \cdot 3 \cdot 5^{4} \cdot 7^{3} \cdot 19^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{7620530425}{526848}$$ = $$-1 \cdot 2^{-9} \cdot 3^{-1} \cdot 5^{2} \cdot 7^{-3} \cdot 673^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.8575369095442787645002575236\dots$$ Stable Faltings height: $$-0.15116188418364159037117597009\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.67144180902244104032824765632\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.38637330886328470276136341575\dots$$ 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: $$54$$  = $$3^{2}\cdot1\cdot1\cdot3\cdot2$$ 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)

## Modular invariants

Modular form 379050.2.a.hv

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4478976 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$14.009068446902112609544136768619307721$$

## 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$$(j)_{-}$$
$$2$$ $$9$$ $$I_{9}$$ Split multiplicative -1 1 9 9
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$5$$ $$1$$ $$IV$$ Additive -1 2 4 0
$$7$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$19$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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
$$3$$ B

## $p$-adic data

### $p$-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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 379050.hv consists of 2 curves linked by isogenies of degree 3.

## 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 $K$ $2$ $$\Q(\sqrt{57})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.4200.1 $$\Z/2\Z$$ Not in database $6$ 6.0.2963520000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.253135681875.3 $$\Z/3\Z$$ Not in database $6$ 6.2.362978280000.2 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.6.673531384135326319516541401783356277587890625.2 $$\Z/9\Z$$ Not in database $18$ 18.0.4502262097326054179041323663501504000000000000.1 $$\Z/6\Z$$ 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.