# Properties

 Label 338.e1 Conductor $338$ Discriminant $-1.336\times 10^{13}$ j-invariant $$-\frac{38575685889}{16384}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2-65773x-6478507$$ y^2+xy+y=x^3-x^2-65773x-6478507 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-65773xz^2-6478507z^3$$ y^2z+xyz+yz^2=x^3-x^2z-65773xz^2-6478507z^3 (dehomogenize, simplify) $$y^2=x^3-1052363x-415676794$$ y^2=x^3-1052363x-415676794 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 1, -65773, -6478507])

gp: E = ellinit([1, -1, 1, -65773, -6478507])

magma: E := EllipticCurve([1, -1, 1, -65773, -6478507]);

oscar: E = EllipticCurve([1, -1, 1, -65773, -6478507])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

trivial

magma: MordellWeilGroup(E);

## Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$338$$ = $2 \cdot 13^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-13364932132864$ = $-1 \cdot 2^{14} \cdot 13^{8}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{38575685889}{16384}$$ = $-1 \cdot 2^{-14} \cdot 3^{3} \cdot 13 \cdot 479^{3}$ 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: $1.4794018384912357915921336337\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.23056439981645536577685799401\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0854699231084624\dots$ Szpiro ratio: $7.710091476905834\dots$

## 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: $0.14905518838523197136213267069\dots$ 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: $14$  = $( 2 \cdot 7 )\cdot1$ 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: $L(E,1)$ ≈ $2.0867726373932475990698573896$ 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

$\displaystyle 2.086772637 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.149055 \cdot 1.000000 \cdot 14}{1^2} \approx 2.086772637$

# 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

$$q + q^{2} + q^{4} - q^{5} + 4 q^{7} + q^{8} - 3 q^{9} - q^{10} + 4 q^{11} + 4 q^{14} + q^{16} + 3 q^{17} - 3 q^{18} + O(q^{20})$$

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: 1092
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 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $14$ $I_{14}$ Split multiplicative -1 1 14 14
$13$ $1$ $IV^{*}$ Additive 1 2 8 0

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
$2$ 2G 4.16.0.2
$7$ 7B.2.3 7.16.0.2

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, 0, 196, 1], [1, 208, 336, 1], [57, 77, 329, 106], [197, 0, 0, 85], [15, 7, 336, 169], [361, 356, 224, 289], [17, 24, 28, 113], [169, 168, 196, 169]]

GL(2,Integers(364)).subgroup(gens)

Gens := [[1, 0, 196, 1], [1, 208, 336, 1], [57, 77, 329, 106], [197, 0, 0, 85], [15, 7, 336, 169], [361, 356, 224, 289], [17, 24, 28, 113], [169, 168, 196, 169]];

sub<GL(2,Integers(364))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$364 = 2^{2} \cdot 7 \cdot 13$$, index $768$, genus $21$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 196 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 208 \\ 336 & 1 \end{array}\right),\left(\begin{array}{rr} 57 & 77 \\ 329 & 106 \end{array}\right),\left(\begin{array}{rr} 197 & 0 \\ 0 & 85 \end{array}\right),\left(\begin{array}{rr} 15 & 7 \\ 336 & 169 \end{array}\right),\left(\begin{array}{rr} 361 & 356 \\ 224 & 289 \end{array}\right),\left(\begin{array}{rr} 17 & 24 \\ 28 & 113 \end{array}\right),\left(\begin{array}{rr} 169 & 168 \\ 196 & 169 \end{array}\right)$.

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

The torsion field $K:=\Q(E[364])$ is a degree-$6604416$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/364\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 338.e consists of 2 curves linked by isogenies of degree 7.

## Twists

The minimal quadratic twist of this elliptic curve is 338.c1, its twist by $13$.

## 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 $K$ $3$ 3.1.676.1 $$\Z/4\Z$$ Not in database $6$ 6.0.1827904.2 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $6$ 6.0.480024727.1 $$\Z/7\Z$$ Not in database $8$ 8.2.999406512.1 $$\Z/3\Z$$ Not in database $12$ 12.2.13685690504052736.19 $$\Z/8\Z$$ Not in database $18$ 18.0.453054841581020940100929875968.2 $$\Z/28\Z$$ Not in database $21$ 21.3.2199205295571527101158910284971023.2 $$\Z/7\Z$$ Not in database

We only show fields where the torsion growth is primitive.

## 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 split ss ord ord ord add ord ss ord ord ord ord ord ord ord 6 0,0 0 0 0 - 0 0,0 0 0 0 0 0 0 0 0 0,0 0 1 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

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