Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
\(y^2+xy=x^3+x^2-1618x+24388\)
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(homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-1618xz^2+24388z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-2097603x+1169307198\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = |
\(\left(16, 46\right)\)
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$\hat{h}(P)$ | ≈ | $0.31164954389922317457019348119$ |
Torsion generators
\( \left(\frac{91}{4}, -\frac{91}{8}\right) \)
Integral points
\( \left(-11, 208\right) \), \( \left(-11, -197\right) \), \( \left(16, 46\right) \), \( \left(16, -62\right) \), \( \left(23, -10\right) \), \( \left(23, -13\right) \), \( \left(24, -2\right) \), \( \left(24, -22\right) \), \( \left(84, 658\right) \), \( \left(84, -742\right) \)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 570 \) | = | $2 \cdot 3 \cdot 5 \cdot 19$ |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | $5540400 $ | = | $2^{4} \cdot 3^{6} \cdot 5^{2} \cdot 19 $ |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{468898230633769}{5540400} \) | = | $2^{-4} \cdot 3^{-6} \cdot 5^{-2} \cdot 19^{-1} \cdot 77689^{3}$ |
Endomorphism ring: | $\Z$ | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | $0.44399078785278025126495506941\dots$ | ||
Stable Faltings height: | $0.44399078785278025126495506941\dots$ |
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: | $0.31164954389922317457019348119\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | $2.1867251700822314888159175386\dots$ | ||
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\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | $2$ | ||
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) $ ≈ $ 1.3629838037781573305271615791 $ |
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: | 384 | ||
$ \Gamma_0(N) $-optimal: | no | ||
Manin constant: | 1 |
Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$3$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$19$ | $1$ | $I_{1}$ | Non-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 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | nonsplit | nonsplit | ord | ord | ss | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 5 | 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 |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 570.b
consists of 2 curves linked by isogenies of
degree 2.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{19}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.0.17100.4 | \(\Z/4\Z\) | Not in database |
$8$ | 8.0.1688960160000.14 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | Deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.2.2850120270000.8 | \(\Z/6\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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.