# Properties

 Label 2448.p5 Conductor $2448$ Discriminant $1.053\times 10^{12}$ j-invariant $$\frac{4354703137}{352512}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3-4899x+122402$$ y^2=x^3-4899x+122402 (homogenize, simplify) $$y^2z=x^3-4899xz^2+122402z^3$$ y^2z=x^3-4899xz^2+122402z^3 (dehomogenize, simplify) $$y^2=x^3-4899x+122402$$ y^2=x^3-4899x+122402 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 0, -4899, 122402])

gp: E = ellinit([0, 0, 0, -4899, 122402])

magma: E := EllipticCurve([0, 0, 0, -4899, 122402]);

oscar: E = EllipticCurve([0, 0, 0, -4899, 122402])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(31, 18\right)$$ (31, 18) $\hat{h}(P)$ ≈ $2.0085874455811399507919527977$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(49, 0\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$(-79,\pm 128)$$, $$(31,\pm 18)$$, $$\left(49, 0\right)$$, $$(3074,\pm 170390)$$

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: $1052595191808$ = $2^{20} \cdot 3^{10} \cdot 17$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{4354703137}{352512}$$ = $2^{-8} \cdot 3^{-4} \cdot 17^{-1} \cdot 23^{3} \cdot 71^{3}$ 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: $1.0498108396955544452268336144\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.19264248519844570988802112552\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $2.0085874455811399507919527977\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.85429244992608963045625439179\dots$ 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: $8$  = $2^{2}\cdot2\cdot1$ 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: $2$ 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: $L'(E,1)$ ≈ $3.4318421795525965639822134408$ 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

$\displaystyle 3.431842180 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.854292 \cdot 2.008587 \cdot 8}{2^2} \approx 3.431842180$

# 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);

Modular degree: 3072
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 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $4$ $I_{12}^{*}$ Additive -1 4 20 8
$3$ $2$ $I_{4}^{*}$ Additive -1 2 10 4
$17$ $1$ $I_{1}$ Non-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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 16.48.0.101

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 = [[5, 4, 812, 813], [431, 528, 582, 593], [1, 288, 612, 613], [1, 16, 0, 1], [271, 0, 0, 815], [15, 2, 718, 803], [568, 273, 543, 10], [1, 0, 16, 1], [801, 16, 800, 17]]

GL(2,Integers(816)).subgroup(gens)

Gens := [[5, 4, 812, 813], [431, 528, 582, 593], [1, 288, 612, 613], [1, 16, 0, 1], [271, 0, 0, 815], [15, 2, 718, 803], [568, 273, 543, 10], [1, 0, 16, 1], [801, 16, 800, 17]];

sub<GL(2,Integers(816))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$816 = 2^{4} \cdot 3 \cdot 17$$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 5 & 4 \\ 812 & 813 \end{array}\right),\left(\begin{array}{rr} 431 & 528 \\ 582 & 593 \end{array}\right),\left(\begin{array}{rr} 1 & 288 \\ 612 & 613 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 271 & 0 \\ 0 & 815 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 718 & 803 \end{array}\right),\left(\begin{array}{rr} 568 & 273 \\ 543 & 10 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 801 & 16 \\ 800 & 17 \end{array}\right)$.

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

The torsion field $K:=\Q(E[816])$ is a degree-$481296384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/816\Z)$.

## Isogenies

gp: ellisomat(E)

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

## Twists

The minimal quadratic twist of this elliptic curve is 102.c5, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{17})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{3})$$ $$\Z/8\Z$$ 2.2.12.1-1734.1-e4 $2$ $$\Q(\sqrt{51})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{17})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.31282289424.2 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.500516630784.16 $$\Z/8\Z$$ Not in database $8$ 8.0.1534132224.11 $$\Z/16\Z$$ Not in database $8$ 8.2.15150586457088.13 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/24\Z$$ 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 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

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