Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-19441587x+32994813166\)
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(homogenize, simplify) |
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\(y^2z=x^3-19441587xz^2+32994813166z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-19441587x+32994813166\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(2546, 0\right) \)
Integral points
\( \left(2546, 0\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 8280 \) | = | $2^{3} \cdot 3^{2} \cdot 5 \cdot 23$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $21151143947520000 $ | = | $2^{11} \cdot 3^{10} \cdot 5^{4} \cdot 23^{4} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{544328872410114151778}{14166950625} \) | = | $2 \cdot 3^{-4} \cdot 5^{-4} \cdot 11^{3} \cdot 23^{-4} \cdot 589139^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $2.6478773948488434861904384811\dots$ | ||
| Stable Faltings height: | $1.4631863350015054401936864180\dots$ | ||
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: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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| Real period: | $0.27888572273083059912890674745\dots$ | ||
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: | $ 32 $ = $ 1\cdot2\cdot2^{2}\cdot2^{2} $ | ||
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) $ ≈ $ 2.2310857818466447930312539796 $ | ||
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: | 196608 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | Additive | 1 | 3 | 11 | 0 |
| $3$ | $2$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
| $5$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
| $23$ | $4$ | $I_{4}$ | 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 | 8.12.0.15 |
The image of the adelic Galois representation has level $552$, index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 192 \\ 204 & 217 \end{array}\right),\left(\begin{array}{rr} 256 & 537 \\ 291 & 94 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 546 & 547 \end{array}\right),\left(\begin{array}{rr} 307 & 300 \\ 162 & 397 \end{array}\right),\left(\begin{array}{rr} 367 & 0 \\ 0 & 551 \end{array}\right),\left(\begin{array}{rr} 545 & 8 \\ 544 & 9 \end{array}\right)$
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 23 |
|---|---|---|---|---|
| Reduction type | add | add | split | split |
| $\lambda$-invariant(s) | - | - | 3 | 1 |
| $\mu$-invariant(s) | - | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
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 and 4.
Its isogeny class 8280.r
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{3})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.0.339738624.5 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.8.95072796278784.5 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.0.92844527616.7 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.2.31726715921280000.1 | \(\Z/6\Z\) | Not in database |
| $16$ | 16.0.1846757322198614016.5 | \(\Z/4\Z \oplus \Z/4\Z\) | 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.