# Properties

 Label 97020.w1 Conductor $97020$ Discriminant $-1.326\times 10^{15}$ j-invariant $$-\frac{3704530032}{33275}$$ CM no Rank $1$ Torsion structure $$\Z/{3}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -110103, 14170702])

gp: E = ellinit([0, 0, 0, -110103, 14170702])

magma: E := EllipticCurve([0, 0, 0, -110103, 14170702]);

$$y^2=x^3-110103x+14170702$$

## Mordell-Weil group structure

$$\Z\times \Z/{3}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(191, 330\right)$$ $$\hat{h}(P)$$ ≈ $0.95735286617541693180171476418$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(147, 1078\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-381,\pm 902)$$, $$(-49,\pm 4410)$$, $$(147,\pm 1078)$$, $$(191,\pm 330)$$, $$(686,\pm 16170)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$97020$$ = $$2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-1325885782636800$$ = $$-1 \cdot 2^{8} \cdot 3^{3} \cdot 5^{2} \cdot 7^{8} \cdot 11^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{3704530032}{33275}$$ = $$-1 \cdot 2^{4} \cdot 3^{3} \cdot 5^{-2} \cdot 7 \cdot 11^{-3} \cdot 107^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.7250295826076098079650854025\dots$$ Stable Faltings height: $$-0.30899504263625669123211581667\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.95735286617541693180171476418\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.48473156798732510116357655687\dots$$ 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: $$108$$  = $$3\cdot2\cdot2\cdot3\cdot3$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$3$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 97020.2.a.w

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);

$$q - q^{5} + q^{11} - 4q^{13} - 3q^{17} + 5q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 580608 $$\Gamma_0(N)$$-optimal: yes 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);

$$L'(E,1)$$ ≈ $$5.5687098712604359401175298687062046968$$

## 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$$(j)_{-}$$
$$2$$ $$3$$ $$IV^{*}$$ Additive -1 2 8 0
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$3$$ $$IV^{*}$$ Additive 1 2 8 0
$$11$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3

## 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.1.1

## $p$-adic data

### $p$-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 add add nonsplit add split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary - - 1 - 2 1 1 1 1 1 1 1 1 1 1 - - 0 - 0 0 0 0 0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 97020.w 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.6468.1 $$\Z/6\Z$$ Not in database $6$ 6.0.5522223168.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.0.52509870000.2 $$\Z/3\Z \times \Z/3\Z$$ Not in database $9$ 9.3.119625333944628508920000.3 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.65662725540866662374926715648000000000000.1 $$\Z/3\Z \times \Z/6\Z$$ 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.