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Properties

Label 24.a4
Conductor $24$
Discriminant $2304$
j-invariant \( \frac{35152}{9} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z \oplus \Z/{4}\Z\)

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

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-x^2-4x+4\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-x^2z-4xz^2+4z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-351x+1890\) Copy content Toggle raw display (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

\( \left(1, 0\right) \), \( \left(4, 6\right) \) Copy content Toggle raw display

comment: Torsion subgroup
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 
oscar: torsion_structure(E)
 

Integral points

\( \left(-2, 0\right) \), \((0,\pm 2)\), \( \left(1, 0\right) \), \( \left(2, 0\right) \), \((4,\pm 6)\) Copy content Toggle raw display

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)[1]
 
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[1]
 
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[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
oscar: tamagawa_numbers(E)
 
Torsion order: $8$
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) $ ≈ $ 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("anayltic 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   24.2.a.a

\( q - q^{3} - 2 q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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: 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)[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 8.96.0.42

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 level \( 24 = 2^{3} \cdot 3 \), index $192$, genus $1$, and generators

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

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

$p$-adic regulators

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

Iwasawa invariants

$p$ 2 3
Reduction type add nonsplit
$\lambda$-invariant(s) - 0
$\mu$-invariant(s) - 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.

Isogenies

gp: ellisomat(E)
 

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

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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$4$ \(\Q(\zeta_{12})\) \(\Z/4\Z \oplus \Z/4\Z\) 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.

Additional information

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