# Properties

 Label 4620n1 Conductor $4620$ Discriminant $-2.376\times 10^{12}$ j-invariant $$-\frac{4890195460096}{9282994875}$$ CM no Rank $1$ Torsion structure $$\Z/{3}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3+x^2-2245x+83975$$ y^2=x^3+x^2-2245x+83975 (homogenize, simplify) $$y^2z=x^3+x^2z-2245xz^2+83975z^3$$ y^2z=x^3+x^2z-2245xz^2+83975z^3 (dehomogenize, simplify) $$y^2=x^3-181872x+61763364$$ y^2=x^3-181872x+61763364 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 1, 0, -2245, 83975])

gp: E = ellinit([0, 1, 0, -2245, 83975])

magma: E := EllipticCurve([0, 1, 0, -2245, 83975]);

oscar: E = EllipticCurve([0, 1, 0, -2245, 83975])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z \oplus \Z/{3}\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(50, 315\right)$$ (50, 315) $\hat{h}(P)$ ≈ $0.23832444413787705089534664050$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(5, 270\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$(-55,\pm 210)$$, $$(-22,\pm 351)$$, $$(5,\pm 270)$$, $$(25,\pm 210)$$, $$(29,\pm 210)$$, $$(50,\pm 315)$$, $$(113,\pm 1134)$$, $$(365,\pm 6930)$$, $$(545,\pm 12690)$$, $$(10970,\pm 1149015)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$4620$$ = $2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-2376446688000$ = $-1 \cdot 2^{8} \cdot 3^{9} \cdot 5^{3} \cdot 7^{3} \cdot 11$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{4890195460096}{9282994875}$$ = $-1 \cdot 2^{16} \cdot 3^{-9} \cdot 5^{-3} \cdot 7^{-3} \cdot 11^{-1} \cdot 421^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.0645077817841732981821979203\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.60240966141087642523737650600\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.23832444413787705089534664050\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.72872311336521791304126503295\dots$ comment: Real Period  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); Tamagawa product: $243$  = $3\cdot3^{2}\cdot3\cdot3\cdot1$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $3$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ (exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $4.6891583349260954900707290932$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

$\displaystyle 4.689158335 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.728723 \cdot 0.238324 \cdot 243}{3^2} \approx 4.689158335$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("anayltic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

## Modular invariants

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

Modular degree: 7776
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: yes
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $3$ $IV^{*}$ Additive -1 2 8 0
$3$ $9$ $I_{9}$ Split multiplicative -1 1 9 9
$5$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$11$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

## Galois representations

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
$3$ 3B.1.1 3.8.0.1

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[4, 3, 9, 7], [2305, 6, 2304, 7], [661, 6, 1983, 19], [1387, 6, 1851, 19], [211, 6, 633, 19], [1, 6, 0, 1], [3, 4, 8, 11], [1, 0, 6, 1], [1926, 391, 1543, 1944]]

GL(2,Integers(2310)).subgroup(gens)

Gens := [[4, 3, 9, 7], [2305, 6, 2304, 7], [661, 6, 1983, 19], [1387, 6, 1851, 19], [211, 6, 633, 19], [1, 6, 0, 1], [3, 4, 8, 11], [1, 0, 6, 1], [1926, 391, 1543, 1944]];

sub<GL(2,Integers(2310))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11$$, index $16$, genus $0$, and generators

$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 2305 & 6 \\ 2304 & 7 \end{array}\right),\left(\begin{array}{rr} 661 & 6 \\ 1983 & 19 \end{array}\right),\left(\begin{array}{rr} 1387 & 6 \\ 1851 & 19 \end{array}\right),\left(\begin{array}{rr} 211 & 6 \\ 633 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 1926 & 391 \\ 1543 & 1944 \end{array}\right)$.

The torsion field $K:=\Q(E[2310])$ is a degree-$1108980$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2310\Z)$.

## $p$-adic regulators

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 split split split nonsplit ord ord ord ord ord ord ord ord ord ord - 6 2 2 1 1 1 1 3 1 3 1 1 3 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

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 4620n consists of 2 curves linked by isogenies of degree 3.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.4620.1 $$\Z/6\Z$$ Not in database $6$ 6.0.24652782000.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $6$ 6.0.6324912.1 $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $9$ 9.3.288178803000000.7 $$\Z/9\Z$$ Not in database $12$ deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.56280390065562432115862208000000.1 $$\Z/3\Z \oplus \Z/6\Z$$ 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.