# Properties

 Label 14586.e4 Conductor $14586$ Discriminant $-2.900\times 10^{18}$ j-invariant $$\frac{2773679829880629422375}{2899504554614368272}$$ CM no Rank $1$ Torsion structure $$\Z/{6}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3+292714x+54762104$$ y^2+xy+y=x^3+292714x+54762104 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3+292714xz^2+54762104z^3$$ y^2z+xyz+yz^2=x^3+292714xz^2+54762104z^3 (dehomogenize, simplify) $$y^2=x^3+379357965x+2553842661966$$ y^2=x^3+379357965x+2553842661966 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 1, 292714, 54762104])

gp: E = ellinit([1, 0, 1, 292714, 54762104])

magma: E := EllipticCurve([1, 0, 1, 292714, 54762104]);

oscar: E = EllipticCurve([1, 0, 1, 292714, 54762104])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z \oplus \Z/{6}\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(-63, 6037\right)$$ (-63, 6037) $\hat{h}(P)$ ≈ $0.93211303552130654193357309234$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(1224, 46792\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-63, 6037\right)$$, $$\left(-63, -5975\right)$$, $$\left(102, 9205\right)$$, $$\left(102, -9308\right)$$, $$\left(234, 11548\right)$$, $$\left(234, -11783\right)$$, $$\left(561, 19609\right)$$, $$\left(561, -20171\right)$$, $$\left(1224, 46792\right)$$, $$\left(1224, -48017\right)$$, $$\left(3369, 196513\right)$$, $$\left(3369, -199883\right)$$, $$\left(8517, 783385\right)$$, $$\left(8517, -791903\right)$$, $$\left(24786, 3890764\right)$$, $$\left(24786, -3915551\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$14586$$ = $2 \cdot 3 \cdot 11 \cdot 13 \cdot 17$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-2899504554614368272$ = $-1 \cdot 2^{4} \cdot 3^{6} \cdot 11^{6} \cdot 13^{4} \cdot 17^{3}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{2773679829880629422375}{2899504554614368272}$$ = $2^{-4} \cdot 3^{-6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{-6} \cdot 13^{-4} \cdot 17^{-3} \cdot 229^{3} \cdot 1753^{3}$ 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: $2.2293332654650940733062542130\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $2.2293332654650940733062542130\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.93211303552130654193357309234\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.16808338249122457901756708971\dots$ 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: $864$  = $2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2^{2}\cdot3$ 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: $6$ 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: $L'(E,1)$ ≈ $3.7601450849900200823438435907$ 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

$\displaystyle 3.760145085 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.168083 \cdot 0.932113 \cdot 864}{6^2} \approx 3.760145085$

# 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

Modular form 14586.2.a.e

$$q - q^{2} + q^{3} + q^{4} - q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{11} + q^{12} + q^{13} + 4 q^{14} + q^{16} + q^{17} - q^{18} + 2 q^{19} + O(q^{20})$$

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: 304128
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is semistable. There are 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$3$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$11$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$13$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$17$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

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
$2$ 2B 2.3.0.1
$3$ 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 = [[2058, 1321, 2057, 1310], [1594, 3, 633, 2236], [409, 12, 210, 73], [1, 0, 12, 1], [1, 6, 6, 37], [11, 2, 2194, 2235], [1505, 2, 1500, 1], [1, 12, 0, 1], [2233, 12, 2232, 13]]

GL(2,Integers(2244)).subgroup(gens)

Gens := [[2058, 1321, 2057, 1310], [1594, 3, 633, 2236], [409, 12, 210, 73], [1, 0, 12, 1], [1, 6, 6, 37], [11, 2, 2194, 2235], [1505, 2, 1500, 1], [1, 12, 0, 1], [2233, 12, 2232, 13]];

sub<GL(2,Integers(2244))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$2244 = 2^{2} \cdot 3 \cdot 11 \cdot 17$$, index $96$, genus $1$, and generators

$\left(\begin{array}{rr} 2058 & 1321 \\ 2057 & 1310 \end{array}\right),\left(\begin{array}{rr} 1594 & 3 \\ 633 & 2236 \end{array}\right),\left(\begin{array}{rr} 409 & 12 \\ 210 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 2194 & 2235 \end{array}\right),\left(\begin{array}{rr} 1505 & 2 \\ 1500 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2233 & 12 \\ 2232 & 13 \end{array}\right)$.

The torsion field $K:=\Q(E[2244])$ is a degree-$2872340$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2244\Z)$.

## $p$-adic regulators

$p$-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 nonsplit split ss ord split split split ord ss ord ord ord ord ord ord 7 4 1,1 1 2 2 2 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

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 14586.e consists of 4 curves linked by isogenies of degrees dividing 6.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-17})$$ $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $4$ 4.2.74052.4 $$\Z/12\Z$$ Not in database $6$ 6.0.12338352.2 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $8$ 8.0.25356622807296.4 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $9$ 9.3.280389568253455762358173488.1 $$\Z/18\Z$$ Not in database $12$ deg 12 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/24\Z$$ Not in database $18$ 18.0.6328348786001025039997966997346099142744003920746925678133248.1 $$\Z/2\Z \oplus \Z/18\Z$$ 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.