# Properties

 Label 30446.i1 Conductor 30446 Discriminant -111304730368 j-invariant $$-\frac{6570725617}{111304730368}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -39, -16055]); // or

magma: E := EllipticCurve("30446m1");

sage: E = EllipticCurve([1, 0, 0, -39, -16055]) # or

sage: E = EllipticCurve("30446m1")

gp: E = ellinit([1, 0, 0, -39, -16055]) \\ or

gp: E = ellinit("30446m1")

$$y^2 + x y = x^{3} - 39 x - 16055$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(78, 637\right)$$ $$\left(30, 85\right)$$ $$\hat{h}(P)$$ ≈ 0.175760589213 2.57646089814

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(26, 13\right)$$, $$\left(26, -39\right)$$, $$\left(30, 85\right)$$, $$\left(30, -115\right)$$, $$\left(52, 325\right)$$, $$\left(52, -377\right)$$, $$\left(78, 637\right)$$, $$\left(78, -715\right)$$, $$\left(132, 1445\right)$$, $$\left(132, -1577\right)$$, $$\left(286, 4693\right)$$, $$\left(286, -4979\right)$$, $$\left(338, 6045\right)$$, $$\left(338, -6383\right)$$, $$\left(936, 28171\right)$$, $$\left(936, -29107\right)$$, $$\left(1430, 53365\right)$$, $$\left(1430, -54795\right)$$, $$\left(44532, 9375173\right)$$, $$\left(44532, -9419705\right)$$

## 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: $$-111304730368$$ = $$-1 \cdot 2^{8} \cdot 13^{5} \cdot 1171$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{6570725617}{111304730368}$$ = $$-1 \cdot 2^{-8} \cdot 13^{-5} \cdot 1171^{-1} \cdot 1873^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.399893210566$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.480515345206$$ 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: $$40$$  = $$2^{3}\cdot5\cdot1$$ 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.i

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 48000 $$\Gamma_0(N)$$-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])

$$L^{(2)}(E,1)/2!$$ ≈ $$7.68619296482$$

## 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$$(j)_{-}$$
$$2$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$13$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$1171$$ $$1$$ $$I_{1}$$ 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 split ordinary ordinary ordinary ordinary split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split 4 2 4 2 2 3 2 2 2 2 2 2 2 2 2 ? 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.i 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 3.1.15223.1 $$\Z/2\Z$$ Not in database
6 6.0.3527773894567.1 $$\Z/2\Z \times \Z/2\Z$$ 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.