# Properties

 Label 30446.c1 Conductor 30446 Discriminant 3166384 j-invariant $$\frac{6826561273}{3166384}$$ CM no Rank 3 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -39, -59]); // or
magma: E := EllipticCurve("30446a1");
sage: E = EllipticCurve([1, 1, 0, -39, -59]) # or
sage: E = EllipticCurve("30446a1")
gp: E = ellinit([1, 1, 0, -39, -59]) \\ or
gp: E = ellinit("30446a1")

$$y^2 + x y = x^{3} + x^{2} - 39 x - 59$$

## Mordell-Weil group structure

$$\Z^3$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-6, 5\right)$$ $$\left(10, 21\right)$$ $$\left(-\frac{17}{4}, \frac{75}{8}\right)$$ $$\hat{h}(P)$$ ≈ 1.32037619939 0.63284288828 3.32066094027

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$$\left(-6, 5\right)$$, $$\left(-5, 9\right)$$, $$\left(-3, 8\right)$$, $$\left(-2, 5\right)$$, $$\left(7, 5\right)$$, $$\left(9, 16\right)$$, $$\left(10, 21\right)$$, $$\left(18, 65\right)$$, $$\left(30, 149\right)$$, $$\left(49, 320\right)$$, $$\left(82, 705\right)$$, $$\left(114, 1165\right)$$, $$\left(127, 1373\right)$$, $$\left(193, 2592\right)$$, $$\left(2142, 98093\right)$$, $$\left(3585, 212896\right)$$, $$\left(41578, 8457377\right)$$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$30446$$ = $$2 \cdot 13 \cdot 1171$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$3166384$$ = $$2^{4} \cdot 13^{2} \cdot 1171$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{6826561273}{3166384}$$ = $$2^{-4} \cdot 7^{3} \cdot 13^{-2} \cdot 271^{3} \cdot 1171^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$3$$ magma: Regulator(E); sage: E.regulator() Regulator: $$0.673965740391$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$1.99091462729$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$4$$  = $$2\cdot2\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 30446.2.a.c

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - q^{2} - q^{3} + q^{4} - 3q^{5} + q^{6} - 5q^{7} - q^{8} - 2q^{9} + 3q^{10} - 3q^{11} - q^{12} - q^{13} + 5q^{14} + 3q^{15} + q^{16} - 6q^{17} + 2q^{18} - 8q^{19} + O(q^{20})$$

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 9856 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

$$L^{(3)}(E,1)/3!$$ ≈ $$5.36723300334$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$13$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$1171$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

## Galois representations

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

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ .

## $p$-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 1171 nonsplit ordinary ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit 6 3 7 3 3 3 3 3 3 3 3 3 3 3 3 ? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ?

An entry ? indicates that the invariants have not yet been computed.

## Isogenies

This curve has no rational isogenies. Its isogeny class 30446.c consists of this curve only.

## 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 3.3.4684.1 $$\Z/2\Z$$ Not in database
6 6.6.102766285504.1 $$\Z/2\Z \times \Z/2\Z$$ 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.