Properties

 Label 30446g1 Conductor 30446 Discriminant -249413632 j-invariant $$-\frac{22164361129}{249413632}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -59, 774]); // or

magma: E := EllipticCurve("30446g1");

sage: E = EllipticCurve([1, 0, 1, -59, 774]) # or

sage: E = EllipticCurve("30446g1")

gp: E = ellinit([1, 0, 1, -59, 774]) \\ or

gp: E = ellinit("30446g1")

$$y^2 + x y + y = x^{3} - 59 x + 774$$

Mordell-Weil group structure

$$\Z$$

Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(47, 296\right)$$ $$\hat{h}(P)$$ ≈ 1.53165358492

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(47, 296\right)$$, $$\left(47, -344\right)$$

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: $$-249413632$$ = $$-1 \cdot 2^{14} \cdot 13 \cdot 1171$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{22164361129}{249413632}$$ = $$-1 \cdot 2^{-14} \cdot 13^{-1} \cdot 53^{6} \cdot 1171^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1.53165358492$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$1.49146160454$$ 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: $$2$$  = $$2\cdot1\cdot1$$ 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: $$1$$ (exact)

Modular invariants

Modular form 30446.2.a.b

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 16800 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$4.56880502672$$

Local data

This elliptic curve is semistable.

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{14}$$ Non-split multiplicative 1 1 14 14
$$13$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$1171$$ $$1$$ $$I_{1}$$ 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 nonsplit ordinary ordinary ordinary ordinary split ordinary ordinary ordinary ordinary ss ordinary ordinary ss ordinary split 4 1 1 1 5 2 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 have not yet been computed.

Isogenies

This curve has no rational isogenies. Its isogeny class 30446g 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.