Properties

 Label 2448n3 Conductor $2448$ Discriminant $8.080\times 10^{13}$ j-invariant $$\frac{576615941610337}{27060804}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

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Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

 $$y^2=x^3-249699x-48023710$$ y^2=x^3-249699x-48023710 (homogenize, simplify) $$y^2z=x^3-249699xz^2-48023710z^3$$ y^2z=x^3-249699xz^2-48023710z^3 (dehomogenize, simplify) $$y^2=x^3-249699x-48023710$$ y^2=x^3-249699x-48023710 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 0, -249699, -48023710])

gp: E = ellinit([0, 0, 0, -249699, -48023710])

magma: E := EllipticCurve([0, 0, 0, -249699, -48023710]);

oscar: E = EllipticCurve([0, 0, 0, -249699, -48023710])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

Infinite order Mordell-Weil generator and height

 $P$ = $$\left(-\frac{35065}{121}, \frac{29770}{1331}\right)$$ (-35065/121, 29770/1331) $\hat{h}(P)$ ≈ $8.0343497823245598031678111908$

sage: E.gens()

magma: Generators(E);

gp: E.gen

Torsion generators

$$\left(-287, 0\right)$$, $$\left(577, 0\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

Integral points

$$\left(-290, 0\right)$$, $$\left(-287, 0\right)$$, $$\left(577, 0\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

Invariants

 Conductor: $$2448$$ = $2^{4} \cdot 3^{2} \cdot 17$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $80803127771136$ = $2^{14} \cdot 3^{10} \cdot 17^{4}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{576615941610337}{27060804}$$ = $2^{-2} \cdot 3^{-4} \cdot 17^{-4} \cdot 83233^{3}$ 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: $1.7429580202554997546440657359\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.50050469536149959952921099598\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0315574162602832\dots$ Szpiro ratio: $6.266496281540907\dots$

BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $8.0343497823245598031678111908\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.21357311248152240761406359795\dots$ 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: $32$  = $2^{2}\cdot2^{2}\cdot2$ 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: $4$ 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: $L'(E,1)$ ≈ $3.4318421795525965639822134408$ 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

$\displaystyle 3.431842180 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.213573 \cdot 8.034350 \cdot 32}{4^2} \approx 3.431842180$

# 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

$$q + 2 q^{5} - 4 q^{11} - 2 q^{13} - q^{17} - 4 q^{19} + O(q^{20})$$

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);

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

Modular degree: 12288
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 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $4$ $I_{6}^{*}$ Additive -1 4 14 2
$3$ $4$ $I_{4}^{*}$ Additive -1 2 10 4
$17$ $2$ $I_{4}$ 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
$2$ 2Cs 8.48.0.137

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 = [[271, 0, 0, 407], [139, 246, 366, 115], [241, 144, 12, 169], [1, 0, 8, 1], [407, 366, 0, 305], [5, 4, 404, 405], [401, 8, 400, 9], [1, 8, 0, 1]]

GL(2,Integers(408)).subgroup(gens)

Gens := [[271, 0, 0, 407], [139, 246, 366, 115], [241, 144, 12, 169], [1, 0, 8, 1], [407, 366, 0, 305], [5, 4, 404, 405], [401, 8, 400, 9], [1, 8, 0, 1]];

sub<GL(2,Integers(408))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$408 = 2^{3} \cdot 3 \cdot 17$$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 271 & 0 \\ 0 & 407 \end{array}\right),\left(\begin{array}{rr} 139 & 246 \\ 366 & 115 \end{array}\right),\left(\begin{array}{rr} 241 & 144 \\ 12 & 169 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 407 & 366 \\ 0 & 305 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 404 & 405 \end{array}\right),\left(\begin{array}{rr} 401 & 8 \\ 400 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[408])$ is a degree-$30081024$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/408\Z)$.

Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 2448n consists of 6 curves linked by isogenies of degrees dividing 8.

Twists

The minimal quadratic twist of this elliptic curve is 102b3, its twist by $12$.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{3})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{3})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ $$\Q(\zeta_{24})$$ $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.27710263296.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.1731891456.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.1358954496.7 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.15150586457088.13 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.196571825135013064605696.1 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.1846757322198614016.5 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ 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 5 7 11 13 17 19 23 29 31 37 41 43 47 add add ord ss ord ord nonsplit ord ss ord ord ord ord ord ss - - 1 1,1 1 1 1 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

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

$p$-adic regulators

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