# Properties

 Label 369600jn1 Conductor $369600$ Discriminant $-1.699\times 10^{19}$ j-invariant $$-\frac{79028701534867456}{16987307596875}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -894033, -380738313])

gp: E = ellinit([0, -1, 0, -894033, -380738313])

magma: E := EllipticCurve([0, -1, 0, -894033, -380738313]);

$$y^2=x^3-x^2-894033x-380738313$$

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);



## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$369600$$ = $$2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-16987307596875000000$$ = $$-1 \cdot 2^{6} \cdot 3^{5} \cdot 5^{11} \cdot 7^{5} \cdot 11^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{79028701534867456}{16987307596875}$$ = $$-1 \cdot 2^{12} \cdot 3^{-5} \cdot 5^{-5} \cdot 7^{-5} \cdot 11^{-3} \cdot 26821^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.4110576221721075335139385322\dots$$ Stable Faltings height: $$1.2597650756750846915049428049\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.076747844037560650326718554597\dots$$ 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: $$30$$  = $$1\cdot1\cdot2\cdot5\cdot3$$ 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 369600.2.a.jn

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);

$$q - q^{3} + q^{7} + q^{9} + q^{11} - 6q^{13} + 7q^{17} - 5q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 11520000 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$2.3024353211268195098015566378983544010$$

## Local data

This elliptic curve is not semistable. There are 5 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$$(j)_{-}$$
$$2$$ $$1$$ $$II$$ Additive -1 6 6 0
$$3$$ $$1$$ $$I_{5}$$ Non-split multiplicative 1 1 5 5
$$5$$ $$2$$ $$I_5^{*}$$ Additive 1 2 11 5
$$7$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$11$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3

## 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 $$p$$ except those listed.

prime Image of Galois representation
$$5$$ B.4.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

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

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 369600jn consists of 2 curves linked by isogenies of degree 5.

## 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 $K$ $2$ $$\Q(\sqrt{-10})$$ $$\Z/5\Z$$ Not in database $3$ 3.1.4620.1 $$\Z/2\Z$$ Not in database $6$ 6.0.24652782000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.13660416000.5 $$\Z/10\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/10\Z$$ Not in database $16$ Deg 16 $$\Z/15\Z$$ Not in database $20$ 20.4.1505680748169532571648000000000000000.1 $$\Z/5\Z$$ 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.