Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-84x+261\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-84xz^2+261z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-108891x+13818870\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{4}\Z\)
Torsion generators
\( \left(-11, 5\right) \), \( \left(7, 5\right) \)
Integral points
\( \left(-11, 5\right) \), \( \left(3, 5\right) \), \( \left(3, -9\right) \), \( \left(5, -3\right) \), \( \left(7, 5\right) \), \( \left(7, -13\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 42 \) | = | $2 \cdot 3 \cdot 7$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $63504 $ | = | $2^{4} \cdot 3^{4} \cdot 7^{2} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{65597103937}{63504} \) | = | $2^{-4} \cdot 3^{-4} \cdot 7^{-2} \cdot 37^{3} \cdot 109^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $-0.15747239686225214586714500626\dots$ | ||
| Stable Faltings height: | $-0.15747239686225214586714500626\dots$ | ||
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: | $3.4754474574968195117771866301\dots$ | ||
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: | $ 16 $ = $ 2^{2}\cdot2\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $8$ | ||
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) $ ≈ $ 0.86886186437420487794429665753 $ | ||
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: | 8 | ||
| $ \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)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
| $3$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 | 8.96.0.37 |
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 7 |
|---|---|---|---|
| Reduction type | split | nonsplit | nonsplit |
| $\lambda$-invariant(s) | 1 | 0 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 42.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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | 2.2.8.1-882.1-a6 |
| $4$ | \(\Q(i, \sqrt{7})\) | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.0.157351936.1 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.0.2439569664.6 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.4.16647192576.6 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
| $8$ | 8.2.6805279152.1 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | 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.