Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
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\(y^2+xy=x^3-x^2-139692x+20130768\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-139692xz^2+20130768z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2235075x+1286134078\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{6}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-147, 6195\right)\)
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| $\hat{h}(P)$ | ≈ | $3.8310182486072146123330100130$ |
Torsion generators
\( \left(168, 1092\right) \)
Integral points
\( \left(-147, 6195\right) \), \( \left(-147, -6048\right) \), \( \left(168, 1092\right) \), \( \left(168, -1260\right) \), \( \left(216, -108\right) \), \( \left(217, -84\right) \), \( \left(217, -133\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 2394 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 19$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $15450607872 $ | = | $2^{8} \cdot 3^{3} \cdot 7^{6} \cdot 19 $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{11165451838341046875}{572244736} \) | = | $2^{-8} \cdot 3^{3} \cdot 5^{6} \cdot 7^{-6} \cdot 17^{3} \cdot 19^{-1} \cdot 1753^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $1.4280440651510995374857787900\dots$ | ||
| Stable Faltings height: | $1.1533909929840721146369674808\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $3.8310182486072146123330100130\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $0.93125828616633892256942875574\dots$ | ||
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: | $ 24 $ = $ 2\cdot2\cdot( 2 \cdot 3 )\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $6$ | ||
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) $ ≈ $ 2.3784449923132826766216916003 $ | ||
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: | 9216 | ||
| $ \Gamma_0(N) $-optimal: | yes | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
| $3$ | $2$ | $III$ | Additive | 1 | 2 | 3 | 0 |
| $7$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
| $19$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 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 | 2.3.0.1 |
| $3$ | 3B.1.1 | 3.8.0.1 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | ss | split | ord | ord | ss | split | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 7 | - | 1,3 | 2 | 1 | 3 | 1,1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | - | 0,0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 2394b
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $4$ | 4.0.513.1 | \(\Z/12\Z\) | Not in database |
| $6$ | 6.0.4560192432.1 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
| $8$ | 8.0.95004009.1 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $8$ | 8.4.21080517080281344.14 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $9$ | 9.3.3519990143155766448.2 | \(\Z/18\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/24\Z\) | Not in database |
| $18$ | 18.6.92039055683773908947530093189460156430713088.3 | \(\Z/2\Z \oplus \Z/18\Z\) | 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.