Properties

 Label 100800cb1 Conductor 100800 Discriminant -3870720000 j-invariant $$-\frac{5400}{7}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -300, 3600]) # or

sage: E = EllipticCurve("100800cb1")

gp: E = ellinit([0, 0, 0, -300, 3600]) \\ or

gp: E = ellinit("100800cb1")

magma: E := EllipticCurve([0, 0, 0, -300, 3600]); // or

magma: E := EllipticCurve("100800cb1");

$$y^2 = x^{3} - 300 x + 3600$$

Mordell-Weil group structure

$$\Z^2$$

Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(10, -40\right)$$ $$\left(-6, 72\right)$$ $$\hat{h}(P)$$ ≈ 0.4284835387142766 1.1567371072527632

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-20,\pm 40)$$, $$(-6,\pm 72)$$, $$(0,\pm 60)$$, $$(10,\pm 40)$$, $$(21,\pm 81)$$, $$(45,\pm 285)$$, $$(90,\pm 840)$$, $$(490,\pm 10840)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$100800$$ = $$2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-3870720000$$ = $$-1 \cdot 2^{15} \cdot 3^{3} \cdot 5^{4} \cdot 7$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{5400}{7}$$ = $$-1 \cdot 2^{3} \cdot 3^{3} \cdot 5^{2} \cdot 7^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.358859288363$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$1.2595536971$$ 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: $$24$$  = $$2^{2}\cdot2\cdot3\cdot1$$ 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: $$1$$ (rounded)

Modular invariants

Modular form 100800.2.a.s

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);

$$q - q^{7} - 4q^{11} - 5q^{13} + 3q^{17} + 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 46080 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$10.8480610415$$

Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$4$$ $$I_5^{*}$$ Additive 1 6 15 0
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$5$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

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(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 add add add nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary - - - 2 2 2 2 2 2 2 2 2 2 2 2 - - - 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 100800cb 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 3.1.4200.1 $$\Z/2\Z$$ Not in database
6 6.0.2963520000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.