Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-19633x+1054463\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-19633xz^2+1054463z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1590300x+773474400\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(59, 324\right)\) | \(\left(113, 540\right)\) |
$\hat{h}(P)$ | ≈ | $0.31517957190164236763678733628$ | $0.33158748524782235100862887049$ |
Integral points
\((-157,\pm 540)\), \((-67,\pm 1440)\), \((-22,\pm 1215)\), \((17,\pm 852)\), \((43,\pm 540)\), \((59,\pm 324)\), \((74,\pm 111)\), \((77,\pm 72)\), \((83,\pm 60)\), \((98,\pm 285)\), \((113,\pm 540)\), \((203,\pm 2340)\), \((323,\pm 5340)\), \((347,\pm 6012)\), \((707,\pm 18468)\), \((2813,\pm 149040)\), \((3569,\pm 213084)\), \((223058,\pm 105348315)\)
Invariants
Conductor: | \( 31200 \) | = | $2^{5} \cdot 3 \cdot 5^{2} \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1965150720000 $ | = | $-1 \cdot 2^{12} \cdot 3^{10} \cdot 5^{4} \cdot 13 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{326938350400}{767637} \) | = | $-1 \cdot 2^{6} \cdot 3^{-10} \cdot 5^{2} \cdot 13^{-1} \cdot 19^{3} \cdot 31^{3}$ | 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: | $1.2391714702412226012747961578\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.0095449855365771669906442585995\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.979311734035211\dots$ | |||
Szpiro ratio: | $3.98839513457756\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.090415950830134955003547494167\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.83203074568955185083726920631\dots$ | 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: | $ 120 $ = $ 2^{2}\cdot( 2 \cdot 5 )\cdot3\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: | $1$ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 9.0274621189712449562729021491 $ | 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
$\displaystyle 9.027462119 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.832031 \cdot 0.090416 \cdot 120}{1^2} \approx 9.027462119$
Modular invariants
Modular form 31200.2.a.bm
For more coefficients, see the Downloads section to the right.
Modular degree: | 92160 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{3}^{*}$ | Additive | -1 | 5 | 12 | 0 |
$3$ | $10$ | $I_{10}$ | Split multiplicative | -1 | 1 | 10 | 10 |
$5$ | $3$ | $IV$ | Additive | -1 | 2 | 4 | 0 |
$13$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 52.2.0.a.1, level \( 52 = 2^{2} \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 27 & 2 \\ 27 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 51 & 0 \end{array}\right),\left(\begin{array}{rr} 51 & 2 \\ 50 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[52])$ is a degree-$1257984$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/52\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 31200.bm consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 31200.w1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.1300.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.87880000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | 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 | add | split | add | ord | ord | nonsplit | ord | ord | ord | ord | ord | ss | ord | ord | ord |
$\lambda$-invariant(s) | - | 3 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 | 2 | 2 | 2 |
$\mu$-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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.