Frey curve for $3 + 125 = 128$ (factored form: $3 + 5^3 = 2^7$)
Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
\(y^2+xy+y=x^3-334x-2368\)
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(homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-334xz^2-2368z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-432243x-109173042\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Torsion generators
\( \left(-11, 5\right) \), \( \left(21, -11\right) \)
Integral points
\( \left(-11, 5\right) \), \( \left(21, -11\right) \)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 30 \) | = | $2 \cdot 3 \cdot 5$ |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | $9000000 $ | = | $2^{6} \cdot 3^{2} \cdot 5^{6} $ |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{4102915888729}{9000000} \) | = | $2^{-6} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{3} \cdot 2287^{3}$ |
Endomorphism ring: | $\Z$ | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | $0.21759239493637902558062626083\dots$ | ||
Stable Faltings height: | $0.21759239493637902558062626083\dots$ |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | $1.1173160864138321498160760446\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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Tamagawa product: | $ 8 $ = $ 2\cdot2\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | $4$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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Special value: | $ L(E,1) $ ≈ $ 0.55865804320691607490803802232 $ |
Modular invariants
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 12 | ||
$ \Gamma_0(N) $-optimal: | no | ||
Manin constant: | 1 |
Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$3$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$5$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
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 | 8.12.0.1 |
$3$ | 3B.1.2 | 3.8.0.2 |
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
$p$ | 2 | 3 | 5 |
---|---|---|---|
Reduction type | nonsplit | split | nonsplit |
$\lambda$-invariant(s) | 0 | 1 | 0 |
$\mu$-invariant(s) | 0 | 1 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 30a
consists of 8 curves linked by isogenies of
degrees dividing 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:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.0.3.1-300.1-a6 |
$3$ | 3.1.243.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{3}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-2}, \sqrt{-3})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$6$ | 6.0.177147.2 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.207360000.1 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | 8.0.12960000.1 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | Deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | Deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | 12.0.8226356490141696.17 | \(\Z/6\Z \oplus \Z/12\Z\) | Not in database |
$16$ | 16.0.11007531417600000000.1 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$18$ | 18.0.617673396283947000000000000.3 | \(\Z/2\Z \oplus \Z/18\Z\) | 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.
Additional information
This is a Frey curve corresponding to the ABC triple $(3,125,128) = (3,5^3,2^7)$ associated to the largest solution of the $S$-unit equation for $S = \{2,3,5\}$. Because the valuation at $2$ of $ABC$ exceeds $4$, there is a quadratic twist with multiplicative reduction at $2$, so the conductor is a multiple of $2$ but not of $4$.