Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-30732800x+65587209600\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-30732800xz^2+65587209600z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-2489356827x+47805607727946\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Torsion generators
\( \left(3202, 98\right) \)
Integral points
\( \left(3201, 0\right) \), \((3202,\pm 98)\)
Invariants
Conductor: | \( 1680 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1475174400 $ | = | $2^{13} \cdot 3 \cdot 5^{2} \cdot 7^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{783736670177727068275201}{360150} \) | = | $2^{-1} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{-4} \cdot 92198401^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.4875765974555083563702703689\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.7944294168955630469530382474\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.074666741320478\dots$ | |||
Szpiro ratio: | $8.52833874606097\dots$ |
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: | $0.43090287435849486501128292544\dots$ | 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: | $ 16 $ = $ 2\cdot1\cdot2\cdot2^{2} $ | 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: | $4$ | 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: | $4$ = $2^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 1.7236114974339794600451317018 $ | 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
$\displaystyle 1.723611497 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.430903 \cdot 1.000000 \cdot 16}{4^2} \approx 1.723611497$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 49152 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{5}^{*}$ | Additive | -1 | 4 | 13 | 1 |
$3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$7$ | $4$ | $I_{4}$ | 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 | 16.96.0.105 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 503 & 26 \\ 3222 & 491 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 30 & 31 \\ 2089 & 3204 \end{array}\right),\left(\begin{array}{rr} 2116 & 29 \\ 671 & 770 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 2711 & 26 \\ 2118 & 875 \end{array}\right),\left(\begin{array}{rr} 3329 & 32 \\ 3328 & 33 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 798 & 1355 \end{array}\right),\left(\begin{array}{rr} 194 & 3331 \\ 799 & 2590 \end{array}\right)$.
The torsion field $K:=\Q(E[3360])$ is a degree-$23781703680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3360\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 1680.j
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 210.e1, its twist by $-4$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{6}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/8\Z\) | Not in database |
$2$ | \(\Q(\sqrt{3}) \) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{3})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$4$ | 4.4.2508800.1 | \(\Z/16\Z\) | Not in database |
$4$ | 4.4.22579200.4 | \(\Z/16\Z\) | Not in database |
$8$ | 8.0.7644119040000.31 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.7644119040000.32 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.458838245376.25 | \(\Z/16\Z\) | Not in database |
$8$ | 8.8.2039281090560000.17 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$8$ | 8.2.272211166080000.4 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | 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 | \(\Z/32\Z\) | Not in database |
$16$ | deg 16 | \(\Z/32\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | 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 | 7 |
---|---|---|---|---|
Reduction type | add | nonsplit | split | split |
$\lambda$-invariant(s) | - | 0 | 1 | 1 |
$\mu$-invariant(s) | - | 0 | 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.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.