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This is a model for the modular curve $X_0(24)$.

## Simplified equation

 $$y^2=x^3-x^2-4x+4$$ y^2=x^3-x^2-4x+4 (homogenize, simplify) $$y^2z=x^3-x^2z-4xz^2+4z^3$$ y^2z=x^3-x^2z-4xz^2+4z^3 (dehomogenize, simplify) $$y^2=x^3-351x+1890$$ y^2=x^3-351x+1890 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, -1, 0, -4, 4])

gp: E = ellinit([0, -1, 0, -4, 4])

magma: E := EllipticCurve([0, -1, 0, -4, 4]);

oscar: E = EllipticCurve([0, -1, 0, -4, 4])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{2}\Z \oplus \Z/{4}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(1, 0\right)$$, $$\left(4, 6\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-2, 0\right)$$, $$(0,\pm 2)$$, $$\left(1, 0\right)$$, $$\left(2, 0\right)$$, $$(4,\pm 6)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$24$$ = $2^{3} \cdot 3$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E);  oscar: conductor(E) Discriminant: $2304$ = $2^{8} \cdot 3^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{35152}{9}$$ = $2^{4} \cdot 3^{-2} \cdot 13^{3}$ 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: $-0.64535228607985727265901067683\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-1.1074504064531541456038320911\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## 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: $4.3130312949992864708773499976\dots$ comment: Real Period  sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Tamagawa product: $8$  = $2^{2}\cdot2$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $8$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)) Analytic order of Ш: $1$ (exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L(E,1)$ ≈ $0.53912891187491080885966874970$ 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

$\displaystyle 0.539128912 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 4.313031 \cdot 1.000000 \cdot 8}{8^2} \approx 0.539128912$

# 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

$$q - q^{3} - 2 q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})$$

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: 1
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 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $4$ $I_{1}^{*}$ Additive -1 3 8 0
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)

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 8.96.0.42

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, 8, 1], [1, 8, 0, 1], [5, 8, 22, 3], [17, 8, 16, 9], [23, 2, 6, 19], [5, 4, 20, 21], [17, 0, 16, 5]]

GL(2,Integers(24)).subgroup(gens)

Gens := [[1, 0, 8, 1], [1, 8, 0, 1], [5, 8, 22, 3], [17, 8, 16, 9], [23, 2, 6, 19], [5, 4, 20, 21], [17, 0, 16, 5]];

sub<GL(2,Integers(24))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.192.1-24.bu.1.7, level $$24 = 2^{3} \cdot 3$$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 22 & 3 \end{array}\right),\left(\begin{array}{rr} 17 & 8 \\ 16 & 9 \end{array}\right),\left(\begin{array}{rr} 23 & 2 \\ 6 & 19 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 20 & 21 \end{array}\right),\left(\begin{array}{rr} 17 & 0 \\ 16 & 5 \end{array}\right)$.

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

The torsion field $K:=\Q(E)$ is a degree-$384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 24a consists of 6 curves linked by isogenies of degrees dividing 8.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\zeta_{12})$$ $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{3})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $4$ $$\Q(\zeta_{8})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ $$\Q(\zeta_{24})$$ $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.2985984.1 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.181398528.1 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ 16.0.36520347436056576.1 $$\Z/8\Z \oplus \Z/8\Z$$ Not in database $16$ 16.8.2393397489569403764736.2 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ 16.0.364791569817010176.2 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database

We only show fields where the torsion growth is primitive.

## Iwasawa invariants

$p$ Reduction type 2 3 add nonsplit - 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 $0$.

This curve is also a quotient of the genus-3 hyperelliptic curve $$y^2 = x^8 + 14x^4 + 1$$ which has geometric automorphism group $S_4 \times \Z/2\Z$.