Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-3104963616x+66592625380884\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-3104963616xz^2+66592625380884z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-251502052923x+48546778408823178\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(32172, 342\right)\)
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| $\hat{h}(P)$ | ≈ | $2.9919933619782666645178107612$ |
Torsion generators
\( \left(32171, 0\right) \)
Integral points
\( \left(32171, 0\right) \), \((32172,\pm 342)\), \((52620,\pm 6993558)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 377520 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $62321937913144166400 $ | = | $2^{11} \cdot 3^{7} \cdot 5^{2} \cdot 11^{7} \cdot 13^{4} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{912446049969377120252018}{17177299425} \) | = | $2 \cdot 3^{-7} \cdot 5^{-2} \cdot 11^{-1} \cdot 13^{-4} \cdot 47^{3} \cdot 1637927^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $3.7862937703109815045008313510\dots$ | ||
| Stable Faltings height: | $1.9519612183985130321707301173\dots$ | ||
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: | $2.9919933619782666645178107612\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $0.10172825219747958645332707703\dots$ | ||
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: | $ 112 $ = $ 2\cdot7\cdot2\cdot2\cdot2 $ | ||
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) $ ≈ $ 8.5223671484142783638433844459 $ | ||
Modular invariants
Modular form 377520.2.a.dy
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: | 165150720 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | Additive | 1 | 4 | 11 | 0 |
| $3$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
| $13$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
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 | 8.12.0.8 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 377520dy
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{66}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.0.312155962500096.18 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | 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.