# Properties

 Label 7220.b2 Conductor $7220$ Discriminant $-4.246\times 10^{19}$ j-invariant $$\frac{298091207216}{3525390625}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3+x^2+319004x-305630796$$ y^2=x^3+x^2+319004x-305630796 (homogenize, simplify) $$y^2z=x^3+x^2z+319004xz^2-305630796z^3$$ y^2z=x^3+x^2z+319004xz^2-305630796z^3 (dehomogenize, simplify) $$y^2=x^3+25839297x-222882368202$$ y^2=x^3+25839297x-222882368202 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, 319004, -305630796])

gp: E = ellinit([0, 1, 0, 319004, -305630796])

magma: E := EllipticCurve([0, 1, 0, 319004, -305630796]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{5035029020845}{628304356}, \frac{11323419022841566187}{15749076987496}\right)$$ (5035029020845/628304356, 11323419022841566187/15749076987496) $\hat{h}(P)$ ≈ $26.353496018273804748594910026$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(519, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(519, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$7220$$ = $2^{2} \cdot 5 \cdot 19^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-42458907602500000000$ = $-1 \cdot 2^{8} \cdot 5^{10} \cdot 19^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{298091207216}{3525390625}$$ = $2^{4} \cdot 5^{-10} \cdot 11^{3} \cdot 19^{-2} \cdot 241^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.4471665270626211980568858203\dots$ Stable Faltings height: $0.51284891710610409510755069005\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $26.353496018273804748594910026\dots$ 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: $0.099874044508917545824553568101\dots$ 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: $4$  = $1\cdot2\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ 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: $L'(E,1)$ ≈ $2.6320302342946592970256260063$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 172800 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$2$ $1$ $IV^{*}$ Additive -1 2 8 0
$5$ $2$ $I_{10}$ Non-split multiplicative 1 1 10 10
$19$ $2$ $I_{2}^{*}$ Additive -1 2 8 2

## 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.12.0.37
sage: gens = [[1, 0, 8, 1], [1, 8, 0, 1], [17, 6, 0, 1], [21, 8, 2, 17], [33, 8, 32, 9], [11, 4, 25, 1], [39, 32, 38, 23], [3, 8, 10, 27], [5, 8, 8, 37]]

sage: GL(2,Integers(40)).subgroup(gens)

magma: Gens := [[1, 0, 8, 1], [1, 8, 0, 1], [17, 6, 0, 1], [21, 8, 2, 17], [33, 8, 32, 9], [11, 4, 25, 1], [39, 32, 38, 23], [3, 8, 10, 27], [5, 8, 8, 37]];

magma: sub<GL(2,Integers(40))|Gens>;

The image of the adelic Galois representation has level $40$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 8 \\ 2 & 17 \end{array}\right),\left(\begin{array}{rr} 33 & 8 \\ 32 & 9 \end{array}\right),\left(\begin{array}{rr} 11 & 4 \\ 25 & 1 \end{array}\right),\left(\begin{array}{rr} 39 & 32 \\ 38 & 23 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 8 & 37 \end{array}\right)$

## $p$-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 add ord nonsplit ord ss ord ord add ord ord ord ord ord ord ord - 1 5 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

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 $d=$ 2.
Its isogeny class 7220.b consists of 2 curves linked by isogenies of degree 2.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.2.144400.1 $$\Z/4\Z$$ Not in database $8$ 8.0.333621760000.15 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.854071705600.18 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ deg 8 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ deg 16 $$\Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.