# Properties

 Label 3864c1 Conductor $3864$ Discriminant $-90139392$ j-invariant $$\frac{128000}{352107}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3+x^2+7x+459$$ y^2=x^3+x^2+7x+459 (homogenize, simplify) $$y^2z=x^3+x^2z+7xz^2+459z^3$$ y^2z=x^3+x^2z+7xz^2+459z^3 (dehomogenize, simplify) $$y^2=x^3+540x+332964$$ y^2=x^3+540x+332964 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 1, 0, 7, 459])

gp: E = ellinit([0, 1, 0, 7, 459])

magma: E := EllipticCurve([0, 1, 0, 7, 459]);

oscar: E = EllipticCurve([0, 1, 0, 7, 459])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(-5, 18\right)$$ (-5, 18) $\hat{h}(P)$ ≈ $0.097771173301500254894393723167$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Integral points

$$(-5,\pm 18)$$, $$(-2,\pm 21)$$, $$(7,\pm 30)$$, $$(13,\pm 54)$$, $$(175,\pm 2322)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$3864$$ = $2^{3} \cdot 3 \cdot 7 \cdot 23$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-90139392$ = $-1 \cdot 2^{8} \cdot 3^{7} \cdot 7 \cdot 23$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{128000}{352107}$$ = $2^{10} \cdot 3^{-7} \cdot 5^{3} \cdot 7^{-1} \cdot 23^{-1}$ 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: $0.20535241092484384592542053856\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.25674570944845302701940087575\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.097771173301500254894393723167\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $1.4983314328005271692168486882\dots$ 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: $28$  = $2^{2}\cdot7\cdot1\cdot1$ 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: $1$ 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: $L'(E,1)$ ≈ $4.1018214210239147072335728972$ 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

$\displaystyle 4.101821421 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.498331 \cdot 0.097771 \cdot 28}{1^2} \approx 4.101821421$

# 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

$$q + q^{3} - q^{7} + q^{9} + q^{11} - 6 q^{13} + 7 q^{19} + O(q^{20})$$

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: 1120
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 4 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $4$ $I_{1}^{*}$ Additive 1 3 8 0
$3$ $7$ $I_{7}$ Split multiplicative -1 1 7 7
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$23$ $1$ $I_{1}$ 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 $\ell$.

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 = [[1, 1, 965, 0], [323, 2, 323, 3], [925, 2, 925, 3], [1, 0, 2, 1], [1, 2, 0, 1], [965, 2, 964, 3], [829, 2, 829, 3]]

GL(2,Integers(966)).subgroup(gens)

Gens := [[1, 1, 965, 0], [323, 2, 323, 3], [925, 2, 925, 3], [1, 0, 2, 1], [1, 2, 0, 1], [965, 2, 964, 3], [829, 2, 829, 3]];

sub<GL(2,Integers(966))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 1 \\ 965 & 0 \end{array}\right),\left(\begin{array}{rr} 323 & 2 \\ 323 & 3 \end{array}\right),\left(\begin{array}{rr} 925 & 2 \\ 925 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 965 & 2 \\ 964 & 3 \end{array}\right),\left(\begin{array}{rr} 829 & 2 \\ 829 & 3 \end{array}\right)$.

The torsion field $K:=\Q(E[966])$ is a degree-$77559939072$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/966\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has no rational isogenies. Its isogeny class 3864c consists of this curve only.

## Twists

This elliptic curve is its own minimal quadratic twist.

## 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 $K$ $3$ 3.1.1932.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1802857392.1 $$\Z/2\Z \oplus \Z/2\Z$$ 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.

## 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 ss nonsplit ord ord ss ord split ord ord ord ord ord ord - 2 1,1 1 1 1 1,1 1 2 1 1 1 1 1 1 - 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.

## $p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.