Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-110977x+14192255\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-110977xz^2+14192255z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8989164x+10373121360\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Torsion generators
\( \left(191, 0\right) \), \( \left(193, 0\right) \)
Integral points
\( \left(-385, 0\right) \), \( \left(191, 0\right) \), \( \left(193, 0\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 3264 \) | = | $2^{6} \cdot 3 \cdot 17$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $7093827403776 $ | = | $2^{20} \cdot 3^{4} \cdot 17^{4} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{576615941610337}{27060804} \) | = | $2^{-2} \cdot 3^{-4} \cdot 17^{-4} \cdot 83233^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $1.5402254662014175636550591781\dots$ | ||
| Stable Faltings height: | $0.50050469536149959952921099591\dots$ | ||
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: | $0.70249775913541607774907964702\dots$ | ||
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: | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2^{2} $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $4$ | ||
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) $ ≈ $ 2.8099910365416643109963185881 $ | ||
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: | 12288 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | Additive | -1 | 6 | 20 | 2 |
| $3$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
| $17$ | $4$ | $I_{4}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 8.96.0.14 |
The image of the adelic Galois representation has level $136$, index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 133 & 26 \\ 42 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 105 & 8 \\ 12 & 33 \end{array}\right),\left(\begin{array}{rr} 129 & 8 \\ 128 & 9 \end{array}\right),\left(\begin{array}{rr} 135 & 94 \\ 0 & 33 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 132 & 133 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 17 |
|---|---|---|---|
| Reduction type | add | split | split |
| $\lambda$-invariant(s) | - | 1 | 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 3264.bc
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.0.8.1-5202.5-i7 |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
| $4$ | 4.0.18432.1 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.0.5473632256.1 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | 8.0.1358954496.9 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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.