Properties

 Label 76050.bl1 Conductor $76050$ Discriminant $-2.024\times 10^{13}$ j-invariant $$\frac{13436683}{7776}$$ CM no Rank $1$ Torsion structure trivial

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, 6813, -779])

gp: E = ellinit([1, -1, 0, 6813, -779])

magma: E := EllipticCurve([1, -1, 0, 6813, -779]);

$$y^2+xy=x^3-x^2+6813x-779$$

Mordell-Weil group structure

$$\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(179, 2543\right)$$ $$\hat{h}(P)$$ ≈ $0.43541297997709558558688483674$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(23, 398\right)$$, $$\left(23, -421\right)$$, $$\left(179, 2543\right)$$, $$\left(179, -2722\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$76050$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-20237981868000$$ = $$-1 \cdot 2^{5} \cdot 3^{11} \cdot 5^{3} \cdot 13^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{13436683}{7776}$$ = $$2^{-5} \cdot 3^{-5} \cdot 13^{2} \cdot 43^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.2426821665854778765563235424\dots$$ Stable Faltings height: $$-0.56396657501094764147598472322\dots$$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.43541297997709558558688483674\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.40838483935651778340498547577\dots$$ 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: $$24$$  = $$1\cdot2^{2}\cdot2\cdot3$$ 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 76050.2.a.bl

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);

$$q - q^{2} + q^{4} - q^{8} - 3q^{11} + q^{16} + 6q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 192000 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$4.2675854371605329974728990328737652116$$

Local data

This elliptic curve is not semistable. There are 4 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{5}$$ Non-split multiplicative 1 1 5 5
$$3$$ $$4$$ $$I_5^{*}$$ Additive -1 2 11 5
$$5$$ $$2$$ $$III$$ Additive -1 2 3 0
$$13$$ $$3$$ $$IV$$ Additive 1 2 4 0

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 $$p$$ .

$p$-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 nonsplit add add ss ordinary add ordinary ss ordinary ordinary ordinary ordinary ss ordinary ordinary 4 - - 1,1 1 - 1 1,1 1 5 1 1 1,1 1 3 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 no rational isogenies. Its isogeny class 76050.bl consists of this curve only.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.20280.1 $$\Z/2\Z$$ Not in database $6$ 6.0.49353408000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.1264873866750000.7 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.