Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-28242x+1833916\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-28242xz^2+1833916z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-451875x+116918750\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 450 \) | = | $2 \cdot 3^{2} \cdot 5^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-56953125000 $ | = | $-1 \cdot 2^{3} \cdot 3^{6} \cdot 5^{10} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{349938025}{8} \) | = | $-1 \cdot 2^{-3} \cdot 5^{2} \cdot 241^{3}$ | 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: | $1.1772316487412783459417995235\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.71327275595452681192312253932\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.050775872125697\dots$ | |||
Szpiro ratio: | $6.933645823186681\dots$ |
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: | $1.0312291893722501215199719290\dots$ | 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: | $ 1 $ | 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: | $1$ | 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) $ ≈ $ 1.0312291893722501215199719290 $ | 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
$\displaystyle 1.031229189 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.031229 \cdot 1.000000 \cdot 1}{1^2} \approx 1.031229189$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 900 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$3$ | $1$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$5$ | $1$ | $II^{*}$ | Additive | 1 | 2 | 10 | 0 |
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 |
---|---|---|
$2$ | 2G | 8.2.0.1 |
$3$ | 3B | 3.4.0.1 |
$5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 46 & 105 \\ 45 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 42 \\ 90 & 61 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 60 & 1 \end{array}\right),\left(\begin{array}{rr} 47 & 30 \\ 0 & 71 \end{array}\right),\left(\begin{array}{rr} 79 & 30 \\ 80 & 119 \end{array}\right),\left(\begin{array}{rr} 31 & 90 \\ 15 & 91 \end{array}\right),\left(\begin{array}{rr} 1 & 72 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 41 & 80 \\ 40 & 81 \end{array}\right),\left(\begin{array}{rr} 61 & 90 \\ 45 & 91 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 30 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 36 \\ 60 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$92160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 5 and 15.
Its isogeny class 450d
consists of 4 curves linked by isogenies of
degrees dividing 15.
Twists
The minimal quadratic twist of this elliptic curve is 50a2, its twist by $-15$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{5}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.200.1 | \(\Z/2\Z\) | Not in database |
$4$ | \(\Q(\zeta_{15})^+\) | \(\Z/15\Z\) | Not in database |
$6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.6834375.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.2.200000.1 | \(\Z/6\Z\) | Not in database |
$10$ | 10.0.607500000000.2 | \(\Z/5\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | 12.0.46708681640625.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | 12.4.145800000000000.1 | \(\Z/30\Z\) | Not in database |
$12$ | 12.0.2560000000000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.6.317733228541125000000000000.1 | \(\Z/9\Z\) | Not in database |
$18$ | 18.0.83682825624000000000000000.1 | \(\Z/6\Z\) | Not in database |
$20$ | 20.0.9226406250000000000000000.1 | \(\Z/15\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | add | add | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 2 | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 |
$\mu$-invariant(s) | 0 | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 |
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 $0$.