Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-3700x+67232\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-3700xz^2+67232z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-4795227x+3151161846\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{8}\Z\)
Torsion generators
\( \left(14, 128\right) \)
Integral points
\( \left(-46, 398\right) \), \( \left(-46, -352\right) \), \( \left(14, 128\right) \), \( \left(14, -142\right) \), \( \left(104, 848\right) \), \( \left(104, -952\right) \)
Invariants
Conductor: | \( 930 \) | = | $2 \cdot 3 \cdot 5 \cdot 31$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1271193750000 $ | = | $2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 31 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{5601911201812801}{1271193750000} \) | = | $2^{-4} \cdot 3^{-8} \cdot 5^{-8} \cdot 31^{-1} \cdot 177601^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.0352520519348703797869572516\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.0352520519348703797869572516\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.81094462040509483207370725743\dots$ | comment: Real Period
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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Tamagawa product: | $ 256 $ = $ 2^{2}\cdot2^{3}\cdot2^{3}\cdot1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $8$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 3.2437784816203793282948290297 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 3.243778482 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.810945 \cdot 1.000000 \cdot 256}{8^2} \approx 3.243778482$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 2048 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$3$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$5$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$31$ | $1$ | $I_{1}$ | 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 | 8.48.0.159 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 15 & 2 \\ 7342 & 7427 \end{array}\right),\left(\begin{array}{rr} 6496 & 5 \\ 4515 & 7426 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 7436 & 7437 \end{array}\right),\left(\begin{array}{rr} 2977 & 16 \\ 1496 & 129 \end{array}\right),\left(\begin{array}{rr} 4961 & 16 \\ 2488 & 129 \end{array}\right),\left(\begin{array}{rr} 4658 & 5587 \\ 2815 & 3742 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 1868 & 1989 \end{array}\right),\left(\begin{array}{rr} 7425 & 16 \\ 7424 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[7440])$ is a degree-$2632974336000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7440\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 930.o
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{8}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{31}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$4$ | 4.0.446400.3 | \(\Z/16\Z\) | Not in database |
$8$ | 8.0.58163441238016.5 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.8.11502047705760000.3 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$8$ | 8.0.3064021032960000.29 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$8$ | 8.2.1635989745870000.11 | \(\Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/32\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 31 |
---|---|---|---|---|
Reduction type | split | split | split | nonsplit |
$\lambda$-invariant(s) | 8 | 1 | 1 | 0 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.