# Properties

 Label 10115g3 Conductor $10115$ Discriminant $-3.300\times 10^{14}$ j-invariant $$-\frac{250523582464}{13671875}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, -37955, -2964672])

gp: E = ellinit([0, -1, 1, -37955, -2964672])

magma: E := EllipticCurve([0, -1, 1, -37955, -2964672]);

$$y^2+y=x^3-x^2-37955x-2964672$$

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);



## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$10115$$ = $$5 \cdot 7 \cdot 17^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-330005826171875$$ = $$-1 \cdot 5^{9} \cdot 7 \cdot 17^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{250523582464}{13671875}$$ = $$-1 \cdot 2^{15} \cdot 5^{-9} \cdot 7^{-1} \cdot 197^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.5440687956591127744324352106\dots$$ Stable Faltings height: $$0.12746212363100473430766790166\dots$$

## 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: $$0.17048101672864762387242175580\dots$$ 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: $$9$$  = $$3^{2}\cdot1\cdot1$$ 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: $$1$$ (exact)

## Modular invariants

Modular form 10115.2.a.f

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);

$$q - q^{3} - 2q^{4} + q^{5} - q^{7} - 2q^{9} + 3q^{11} + 2q^{12} + 5q^{13} - q^{15} + 4q^{16} + 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 30240 $$\Gamma_0(N)$$-optimal: no 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);

$$L(E,1)$$ ≈ $$1.5343291505578286148517958021776063018$$

## Local data

This elliptic curve is not semistable. There are 3 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$$(j)_{-}$$
$$5$$ $$9$$ $$I_{9}$$ Split multiplicative -1 1 9 9
$$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$17$$ $$1$$ $$I_0^{*}$$ Additive 1 2 6 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 $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B

## $p$-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 $$0$$.

## 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 ss ordinary split nonsplit ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 0,5 0 1 0 0 0 - 0 0 0 0 0 0 0 0 0,0 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 10115g consists of 3 curves linked by isogenies of degrees dividing 9.

## 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 $K$ $2$ $$\Q(\sqrt{-51})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.140.1 $$\Z/2\Z$$ Not in database $6$ 6.0.686000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.8599366377.7 $$\Z/3\Z$$ Not in database $6$ 6.0.232182892179.5 $$\Z/9\Z$$ Not in database $6$ 6.0.2599959600.3 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.2.1994230763663705399420278464449088000000.1 $$\Z/6\Z$$ Not in database $18$ 18.0.39252444121192713376789341015751399104000000.1 $$\Z/18\Z$$ 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.