# Properties

 Label 12635.a1 Conductor $12635$ Discriminant $108345125$ j-invariant $$\frac{533794816}{300125}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -120, -126])

gp: E = ellinit([0, 1, 1, -120, -126])

magma: E := EllipticCurve([0, 1, 1, -120, -126]);

$$y^2+y=x^3+x^2-120x-126$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-9, 17\right)$$ $$\left(12, 17\right)$$ $$\hat{h}(P)$$ ≈ $0.10720965331083064329530961612$ $1.3898141071429286689485446873$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-9, 17\right)$$, $$\left(-9, -18\right)$$, $$\left(-7, 20\right)$$, $$\left(-7, -21\right)$$, $$\left(-4, 17\right)$$, $$\left(-4, -18\right)$$, $$\left(-2, 10\right)$$, $$\left(-2, -11\right)$$, $$\left(11, 2\right)$$, $$\left(11, -3\right)$$, $$\left(12, 17\right)$$, $$\left(12, -18\right)$$, $$\left(26, 122\right)$$, $$\left(26, -123\right)$$, $$\left(61, 472\right)$$, $$\left(61, -473\right)$$, $$\left(201, 2852\right)$$, $$\left(201, -2853\right)$$, $$\left(376, 7297\right)$$, $$\left(376, -7298\right)$$, $$\left(2786, 147077\right)$$, $$\left(2786, -147078\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$12635$$ = $$5 \cdot 7 \cdot 19^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$108345125$$ = $$5^{3} \cdot 7^{4} \cdot 19^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{533794816}{300125}$$ = $$2^{12} \cdot 5^{-3} \cdot 7^{-4} \cdot 19^{4}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.23185667619506500569786412040\dots$$ Stable Faltings height: $$-0.25888315366600840430364045158\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.14753054814317521429908635908\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$1.5509359059995792263715126587\dots$$ 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: $$12$$  = $$3\cdot2^{2}\cdot1$$ 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 12635.2.a.a

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6912 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$2.7457250921645999008997127246829377619$$

## 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$$(j)_{-}$$
$$5$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$7$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$19$$ $$1$$ $$II$$ Additive -1 2 2 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 $$p$$ .

## $p$-adic data

### $p$-adic regulators

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

Note: $$p$$-adic regulator data only exists for primes $$p\ge 5$$ 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 ss ordinary split split ordinary ss ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary 12,3 2 3 5 2 2,2 2 - 2 2 2 2 2 2 2 0,0 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 no rational isogenies. Its isogeny class 12635.a consists of this curve only.

## 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 $K$ $3$ 3.3.7220.1 $$\Z/2\Z$$ Not in database $6$ 6.6.260642000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.