# Properties

 Label 644.a1 Conductor $644$ Discriminant $-18032$ j-invariant $$\frac{32000}{1127}$$ CM no Rank $1$ Torsion structure trivial

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-x^2+2x-7$$ y^2=x^3-x^2+2x-7 (homogenize, simplify) $$y^2z=x^3-x^2z+2xz^2-7z^3$$ y^2z=x^3-x^2z+2xz^2-7z^3 (dehomogenize, simplify) $$y^2=x^3+135x-4671$$ y^2=x^3+135x-4671 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, 2, -7])

gp: E = ellinit([0, -1, 0, 2, -7])

magma: E := EllipticCurve([0, -1, 0, 2, -7]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(4, 7\right)$$ (4, 7) $\hat{h}(P)$ ≈ $0.16656685583803490401207698963$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(2,\pm 1)$$, $$(4,\pm 7)$$, $$(11,\pm 35)$$, $$(22,\pm 101)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$644$$ = $2^{2} \cdot 7 \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-18032$ = $-1 \cdot 2^{4} \cdot 7^{2} \cdot 23$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{32000}{1127}$$ = $2^{8} \cdot 5^{3} \cdot 7^{-2} \cdot 23^{-1}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.50341192950283168496052130358\dots$ Stable Faltings height: $-0.73446098968948012143293201073\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.16656685583803490401207698963\dots$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $1.8677219380527065717314592326\dots$ 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: $6$  = $3\cdot2\cdot1$ 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) 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); Special value: $L'(E,1)$ ≈ $1.8666034248069619978103485572$

## Modular invariants

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);

$$q - q^{3} + q^{7} - 2 q^{9} - 2 q^{11} - 3 q^{13} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 24 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## 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$(j)_{-}$
$2$ $3$ $IV$ Additive -1 2 4 0
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$23$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$.

sage: gens = [[5, 2, 5, 3], [1, 1, 45, 0], [1, 2, 0, 1], [45, 2, 44, 3], [1, 0, 2, 1]]

sage: GL(2,Integers(46)).subgroup(gens)

magma: Gens := [[5, 2, 5, 3], [1, 1, 45, 0], [1, 2, 0, 1], [45, 2, 44, 3], [1, 0, 2, 1]];

magma: sub<GL(2,Integers(46))|Gens>;

The image of the adelic Galois representation has level $46$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 45 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 45 & 2 \\ 44 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right)$

## $p$-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 add ord ss split ord ord ss ss nonsplit ord ord ord ord ord ord - 1 5,1 2 1 1 1,1 1,1 1 1 1 1 1 1 1 - 0 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 644.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 $K$ $3$ 3.1.23.1 $$\Z/2\Z$$ Not in database $6$ 6.0.12167.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ 8.2.23511063249072.7 $$\Z/3\Z$$ 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.