# Properties

 Label 249690cj2 Conductor $249690$ Discriminant $4.497\times 10^{27}$ j-invariant $$\frac{33716734605576776462761920782881}{4496908623986020832100000000}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \oplus \Z/{8}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3-673039570x-5895575919100$$ y^2+xy=x^3-673039570x-5895575919100 (homogenize, simplify) $$y^2z+xyz=x^3-673039570xz^2-5895575919100z^3$$ y^2z+xyz=x^3-673039570xz^2-5895575919100z^3 (dehomogenize, simplify) $$y^2=x^3-872259282747x-275061373303681386$$ y^2=x^3-872259282747x-275061373303681386 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 0, -673039570, -5895575919100])

gp: E = ellinit([1, 0, 0, -673039570, -5895575919100])

magma: E := EllipticCurve([1, 0, 0, -673039570, -5895575919100]);

oscar: E = EllipticCurve([1, 0, 0, -673039570, -5895575919100])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z \oplus \Z/{8}\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(29720, 579230\right)$$ (29720, 579230) $\hat{h}(P)$ ≈ $4.7456146183018982790402558639$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(-10460, 5230\right)$$, $$\left(37100, 4475870\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-17710, 693980\right)$$, $$\left(-17710, -676270\right)$$, $$\left(-14560, 911330\right)$$, $$\left(-14560, -896770\right)$$, $$\left(-11620, 602630\right)$$, $$\left(-11620, -591010\right)$$, $$\left(-10460, 5230\right)$$, $$\left(29540, -14770\right)$$, $$\left(29720, 579230\right)$$, $$\left(29720, -608950\right)$$, $$\left(37100, 4475870\right)$$, $$\left(37100, -4512970\right)$$, $$\left(73640, 18507230\right)$$, $$\left(73640, -18580870\right)$$, $$\left(212492, 97082846\right)$$, $$\left(212492, -97295338\right)$$, $$\left(286790, 152791730\right)$$, $$\left(286790, -153078520\right)$$, $$\left(408590, 260432480\right)$$, $$\left(408590, -260841070\right)$$, $$\left(10834040, 35654835230\right)$$, $$\left(10834040, -35665669270\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$249690$$ = $2 \cdot 3 \cdot 5 \cdot 7 \cdot 29 \cdot 41$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $4496908623986020832100000000$ = $2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 29^{4} \cdot 41^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{33716734605576776462761920782881}{4496908623986020832100000000}$$ = $2^{-8} \cdot 3^{-8} \cdot 5^{-8} \cdot 7^{-8} \cdot 29^{-4} \cdot 41^{-2} \cdot 409^{3} \cdot 78987529^{3}$ 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: $4.0337723264808702804863965606\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $4.0337723264808702804863965606\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $0.9972836689888865\dots$ Szpiro ratio: $5.8413011438419815\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $4.7456146183018982790402558639\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.029902133792095422678190804484\dots$ 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: $32768$  = $2^{3}\cdot2^{3}\cdot2^{3}\cdot2^{3}\cdot2^{2}\cdot2$ 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: $16$ 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: $1$ ( rounded) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $18.163712414999963320828938200$ 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

$\displaystyle 18.163712415 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.029902 \cdot 4.745615 \cdot 32768}{16^2} \approx 18.163712415$

# 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

Modular form 249690.2.a.cj

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} - 2 q^{13} + q^{14} + q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

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);

For more coefficients, see the Downloads section to the right.

Modular degree: 190840832
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 6 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$3$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$5$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$7$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$29$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$41$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

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
$2$ 2Cs 8.96.0.40

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, 16, 0, 1], [1598017, 8, 0, 1], [682089, 16, 1607954, 345], [15, 16, 224, 1248689], [1, 0, 16, 1], [1997505, 16, 1997504, 17], [9, 8, 1331636, 1997481], [5, 8, 68, 499489], [1722009, 16, 137828, 121], [1, 16, 4, 65], [1426809, 8, 570676, 1997481]]

GL(2,Integers(1997520)).subgroup(gens)

Gens := [[1, 16, 0, 1], [1598017, 8, 0, 1], [682089, 16, 1607954, 345], [15, 16, 224, 1248689], [1, 0, 16, 1], [1997505, 16, 1997504, 17], [9, 8, 1331636, 1997481], [5, 8, 68, 499489], [1722009, 16, 137828, 121], [1, 16, 4, 65], [1426809, 8, 570676, 1997481]];

sub<GL(2,Integers(1997520))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$1997520 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 29 \cdot 41$$, index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1598017 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 682089 & 16 \\ 1607954 & 345 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 224 & 1248689 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1997505 & 16 \\ 1997504 & 17 \end{array}\right),\left(\begin{array}{rr} 9 & 8 \\ 1331636 & 1997481 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 68 & 499489 \end{array}\right),\left(\begin{array}{rr} 1722009 & 16 \\ 137828 & 121 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1426809 & 8 \\ 570676 & 1997481 \end{array}\right)$.

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

The torsion field $K:=\Q(E[1997520])$ is a degree-$2793260409610567680000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1997520\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 249690cj consists of 8 curves linked by isogenies of degrees dividing 16.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{8}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(i, \sqrt{41})$$ $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{-210}, \sqrt{-290})$$ $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $4$ $$\Q(\sqrt{210}, \sqrt{11890})$$ $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/24\Z$$ Not in database $16$ deg 16 $$\Z/8\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/32\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/32\Z$$ 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

No Iwasawa invariant data is available for this curve.

## $p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.