# Properties

 Label 122010do1 Conductor 122010 Discriminant -5744159273484243360 j-invariant $$-\frac{68921624431417353829938395089}{117227740275188640}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -312566640, -2126999407968]) # or

sage: E = EllipticCurve("122010do1")

gp: E = ellinit([1, 0, 0, -312566640, -2126999407968]) \\ or

gp: E = ellinit("122010do1")

magma: E := EllipticCurve([1, 0, 0, -312566640, -2126999407968]); // or

magma: E := EllipticCurve("122010do1");

$$y^2 + x y = x^{3} - 312566640 x - 2126999407968$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{6180124323592025041003092}{77200269818673191281}, -\frac{14978828076200848847033278911192737868}{678310011901066286166592641929}\right)$$ $$\hat{h}(P)$$ ≈ 55.63920528301045

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$122010$$ = $$2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 83$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-5744159273484243360$$ = $$-1 \cdot 2^{5} \cdot 3^{3} \cdot 5 \cdot 7^{2} \cdot 83^{7}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{68921624431417353829938395089}{117227740275188640}$$ = $$-1 \cdot 2^{-5} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{4} \cdot 83^{-7} \cdot 127^{3} \cdot 2410927^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$55.639205283$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.017952870695$$ 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: $$15$$  = $$5\cdot3\cdot1\cdot1\cdot1$$ 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 122010.2.a.dj

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 20885760 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$14.9832518703$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$3$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$5$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$7$$ $$1$$ $$II$$ Additive -1 2 2 0
$$83$$ $$1$$ $$I_{7}$$ Non-split multiplicative 1 1 7 7

## 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
$$7$$ B.1.3

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ 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 83 split split split add ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary nonsplit 7 2 2 - 1 3,1 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 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 $$d=$$ 7.
Its isogeny class 122010do consists of 2 curves linked by isogenies of degree 7.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.488040.1 $$\Z/2\Z$$ Not in database
6 $$\Q(\zeta_{7})$$ $$\Z/7\Z$$ Not in database
$$x^{6} - 686 x^{4} - 1764 x^{3} + 239659 x^{2} + 3289272 x + 15541134$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
7 7.1.12252303000000.9 $$\Z/7\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.