Frey curve for $3 + 125 = 128$ (factored form: $3 + 5^3 = 2^7$)
Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
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\(y^2+xy+y=x^3-334x-2368\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-334xz^2-2368z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-432243x-109173042\)
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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
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 30 \) | = | $2 \cdot 3 \cdot 5$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $9000000 $ | = | $2^{6} \cdot 3^{2} \cdot 5^{6} $ |
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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.
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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 $4$.