# Properties

 Label 1386.k1 Conductor $1386$ Discriminant $448278138$ j-invariant $$\frac{1285429208617}{614922}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2-2039x+35925$$ y^2+xy+y=x^3-x^2-2039x+35925 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-2039xz^2+35925z^3$$ y^2z+xyz+yz^2=x^3-x^2z-2039xz^2+35925z^3 (dehomogenize, simplify) $$y^2=x^3-32619x+2266598$$ y^2=x^3-32619x+2266598 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 1, -2039, 35925])

gp: E = ellinit([1, -1, 1, -2039, 35925])

magma: E := EllipticCurve([1, -1, 1, -2039, 35925]);

oscar: E = EllipticCurve([1, -1, 1, -2039, 35925])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(\frac{107}{4}, -\frac{111}{8}\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$1386$$ = $2 \cdot 3^{2} \cdot 7 \cdot 11$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $448278138$ = $2 \cdot 3^{7} \cdot 7 \cdot 11^{4}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{1285429208617}{614922}$$ = $2^{-1} \cdot 3^{-1} \cdot 7^{-1} \cdot 11^{-4} \cdot 83^{3} \cdot 131^{3}$ 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: $0.61496673976284453028575701520\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.065660595428789684588134396739\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $0.9493319727776763\dots$ Szpiro ratio: $4.765405472995891\dots$

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $1.6456304289591871305477427958\dots$ 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: $8$  = $1\cdot2^{2}\cdot1\cdot2$ 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: $2$ 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: $L(E,1)$ ≈ $3.2912608579183742610954855916$ 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

$\displaystyle 3.291260858 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.645630 \cdot 1.000000 \cdot 8}{2^2} \approx 3.291260858$

# 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("analytic 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

$$q + q^{2} + q^{4} + 2 q^{5} + q^{7} + q^{8} + 2 q^{10} - q^{11} + 2 q^{13} + q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})$$

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: 1024
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 not semistable. There are 4 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$3$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$7$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$11$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4

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 4.6.0.1

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 = [[235, 232, 1178, 237], [1, 0, 8, 1], [1, 8, 0, 1], [1841, 8, 1840, 9], [1, 4, 4, 17], [796, 1, 1343, 6], [7, 6, 1842, 1843], [232, 1163, 697, 726], [1228, 1847, 593, 1842], [673, 8, 844, 33]]

GL(2,Integers(1848)).subgroup(gens)

Gens := [[235, 232, 1178, 237], [1, 0, 8, 1], [1, 8, 0, 1], [1841, 8, 1840, 9], [1, 4, 4, 17], [796, 1, 1343, 6], [7, 6, 1842, 1843], [232, 1163, 697, 726], [1228, 1847, 593, 1842], [673, 8, 844, 33]];

sub<GL(2,Integers(1848))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11$$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 235 & 232 \\ 1178 & 237 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1841 & 8 \\ 1840 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 796 & 1 \\ 1343 & 6 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1842 & 1843 \end{array}\right),\left(\begin{array}{rr} 232 & 1163 \\ 697 & 726 \end{array}\right),\left(\begin{array}{rr} 1228 & 1847 \\ 593 & 1842 \end{array}\right),\left(\begin{array}{rr} 673 & 8 \\ 844 & 33 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[1848])$ is a degree-$40874803200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1848\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 1386.k consists of 4 curves linked by isogenies of degrees dividing 4.

## Twists

The minimal quadratic twist of this elliptic curve is 462.a1, its twist by $-3$.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{42})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{6})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{7})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{6}, \sqrt{7})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.359729184374784.48 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.8.243731546505216.8 $$\Z/8\Z$$ Not in database $8$ 8.0.5143354480889856.10 $$\Z/8\Z$$ Not in database $8$ 8.2.99636092064432.3 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ 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 7 11 split add split nonsplit 3 - 1 0 0 - 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

## $p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.