Properties

 Label 20510.c1 Conductor 20510 Discriminant -12976912684820654990 j-invariant $$-\frac{18864891308791949569351281}{12976912684820654990}$$ CM no Rank 0 Torsion Structure $$\mathrm{Trivial}$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -5545962, -5028652761]); // or

magma: E := EllipticCurve("20510e2");

sage: E = EllipticCurve([1, -1, 1, -5545962, -5028652761]) # or

sage: E = EllipticCurve("20510e2")

gp: E = ellinit([1, -1, 1, -5545962, -5028652761]) \\ or

gp: E = ellinit("20510e2")

$$y^2 + x y + y = x^{3} - x^{2} - 5545962 x - 5028652761$$

Trivial

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

None

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$20510$$ = $$2 \cdot 5 \cdot 7 \cdot 293$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-12976912684820654990$$ = $$-1 \cdot 2 \cdot 5 \cdot 7 \cdot 293^{7}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{18864891308791949569351281}{12976912684820654990}$$ = $$-1 \cdot 2^{-1} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{-1} \cdot 13^{3} \cdot 293^{-7} \cdot 6825799^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$0$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0491878030933$$ 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: $$1$$  = $$1\cdot1\cdot1\cdot1$$ 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: $$49$$ (exact)

Modular invariants

Modular form 20510.2.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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 2765952 $$\Gamma_0(N)$$-optimal: no 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])

$$L(E,1)$$ ≈ $$2.41020235157$$

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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$5$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$7$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$293$$ $$1$$ $$I_{7}$$ Non-split multiplicative 1 1 7 7

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 $$p$$ 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]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 293 split ss split split nonsplit 2 0,0 1 3 0 0 0,0 0 1 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 20510.c 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 3.1.82040.1 $$\Z/2\Z$$ Not in database
6 $$\Q(\zeta_{7})$$ $$\Z/7\Z$$ Not in database
$$x^{6} - 78 x^{4} + 1521 x^{2} + 82040$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
7 $$x^{7} - 2800$$ $$\Z/7\Z$$ 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.