Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+24x-56\) | (homogenize, simplify) |
\(y^2z=x^3+24xz^2-56z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+24x-56\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(2, 0\right) \)
Integral points
\( \left(2, 0\right) \)
Invariants
Conductor: | \( 576 \) | = | $2^{6} \cdot 3^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-2239488 $ | = | $-1 \cdot 2^{10} \cdot 3^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{2048}{3} \) | = | $2^{11} \cdot 3^{-1}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $-0.096046141745802426961388058365\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.2229749365464783638400374447\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.1757189916348774\dots$ | |||
Szpiro ratio: | $3.394191865641939\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.3764094010021856987734438004\dots$ | 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);
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Tamagawa product: | $ 4 $ = $ 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)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 1.3764094010021856987734438004 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 1.376409401 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.376409 \cdot 1.000000 \cdot 4}{2^2} \approx 1.376409401$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 64 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_0^{*}$ | additive | 1 | 6 | 10 | 0 |
$3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 | 16.96.0.268 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 48.192.1-48.dt.1.5, level \( 48 = 2^{4} \cdot 3 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 47 \\ 1 & 38 \end{array}\right),\left(\begin{array}{rr} 45 & 16 \\ 26 & 11 \end{array}\right),\left(\begin{array}{rr} 47 & 32 \\ 12 & 11 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 46 & 35 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 44 & 45 \end{array}\right),\left(\begin{array}{rr} 33 & 16 \\ 32 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[48])$ is a degree-$6144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | additive | $2$ | \( 9 = 3^{2} \) |
$3$ | additive | $8$ | \( 64 = 2^{6} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 576.d
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 24.a5, its twist by $-24$.
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:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.3.1-12288.1-g4 |
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | 2.2.8.1-648.1-a3 |
$2$ | \(\Q(\sqrt{-6}) \) | \(\Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | \(\Q(i, \sqrt{6})\) | \(\Z/8\Z\) | not in database |
$4$ | \(\Q(\sqrt{-2}, \sqrt{3})\) | \(\Z/8\Z\) | not in database |
$8$ | 8.0.47775744.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.4.191102976.1 | \(\Z/8\Z\) | not in database |
$8$ | \(\Q(\zeta_{24})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.2.11609505792.10 | \(\Z/6\Z\) | not in database |
$16$ | 16.0.36520347436056576.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$16$ | 16.0.5258930030792146944.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$16$ | 16.4.5258930030792146944.1 | \(\Z/16\Z\) | not in database |
$16$ | 16.0.7213895789838336.1 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$16$ | 16.0.584325558976905216.6 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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
$p$ | 2 | 3 |
---|---|---|
Reduction type | add | add |
$\lambda$-invariant(s) | - | - |
$\mu$-invariant(s) | - | - |
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 $0$.