# Properties

 Label 10051b3 Conductor $10051$ Discriminant $-2812681891$ j-invariant $$-\frac{50357871050752}{19}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -406977, 99796080])

gp: E = ellinit([0, 1, 1, -406977, 99796080])

magma: E := EllipticCurve([0, 1, 1, -406977, 99796080]);

$$y^2+y=x^3+x^2-406977x+99796080$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(360, 264\right)$$ $$\left(\frac{13213}{36}, \frac{6769}{216}\right)$$ $$\hat{h}(P)$$ ≈ $0.33520833616006797055143310157$ $2.8932249383841783705438480152$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(224, 4463\right)$$, $$\left(224, -4464\right)$$, $$\left(360, 264\right)$$, $$\left(360, -265\right)$$, $$\left(368, 0\right)$$, $$\left(368, -1\right)$$, $$\left(1050, 28830\right)$$, $$\left(1050, -28831\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$10051$$ = $$19 \cdot 23^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-2812681891$$ = $$-1 \cdot 19 \cdot 23^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{50357871050752}{19}$$ = $$-1 \cdot 2^{18} \cdot 19^{-1} \cdot 577^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.6011862649558926489234712995\dots$$ Stable Faltings height: $$0.033439156991317803520094883596\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.94442984883344241395105604338\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.86055825181833166799702650140\dots$$ 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: $$4$$  = $$1\cdot2^{2}$$ 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 10051.2.a.c

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 38016 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$3.2509475987086337902439572489875197408$$

## Local data

This elliptic curve is not semistable. There are 2 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$$(j)_{-}$$
$$19$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$23$$ $$4$$ $$I_0^{*}$$ 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]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

## 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 ordinary ordinary ordinary ordinary ordinary nonsplit add ordinary ordinary ordinary ordinary ordinary ordinary 4,7 2 2 2 2 2 2 4 - 2 2 2 2 2 2 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 10051b 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 $K$ $2$ $$\Q(\sqrt{69})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.76.1 $$\Z/2\Z$$ Not in database $6$ 6.0.109744.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.14270540463.1 $$\Z/3\Z$$ Not in database $6$ 6.6.31209671992581.1 $$\Z/9\Z$$ Not in database $6$ 6.2.1897467984.1 $$\Z/6\Z$$ 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 $18$ 18.0.4297228167484676083994283208121413632.1 $$\Z/6\Z$$ Not in database $18$ 18.6.44950524425770151928627097389761944698379063296.1 $$\Z/18\Z$$ 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.