Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-25932x+1171600\) | (homogenize, simplify) |
\(y^2z=x^3-25932xz^2+1171600z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-25932x+1171600\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(32, 612\right)\) |
$\hat{h}(P)$ | ≈ | $3.2062318295241649522173851559$ |
Torsion generators
\( \left(50, 0\right) \)
Integral points
\((32,\pm 612)\), \( \left(50, 0\right) \), \((1074,\pm 34816)\)
Invariants
Conductor: | \( 6336 \) | = | $2^{6} \cdot 3^{2} \cdot 11$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $523077892964352 $ | = | $2^{28} \cdot 3^{11} \cdot 11 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{10091699281}{2737152} \) | = | $2^{-10} \cdot 3^{-5} \cdot 11^{-1} \cdot 2161^{3}$ | 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: | $1.5321612253534702806383667881\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.056865689820502529185104012548\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.073399617726021\dots$ | |||
Szpiro ratio: | $4.809605659430858\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.2062318295241649522173851559\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.48647471136520843363791015828\dots$ | 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: | $ 8 $ = $ 2^{2}\cdot2\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: | $2$ | 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'(E,1) $ ≈ $ 3.1195014076754246339736250491 $ | 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
$\displaystyle 3.119501408 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.486475 \cdot 3.206232 \cdot 8}{2^2} \approx 3.119501408$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 30720 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{18}^{*}$ | Additive | -1 | 6 | 28 | 10 |
$3$ | $2$ | $I_{5}^{*}$ | Additive | -1 | 2 | 11 | 5 |
$11$ | $1$ | $I_{1}$ | 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 | 8.6.0.4 |
$5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 1301 & 20 \\ 1300 & 21 \end{array}\right),\left(\begin{array}{rr} 659 & 1300 \\ 1310 & 1119 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 736 & 5 \\ 75 & 1306 \end{array}\right),\left(\begin{array}{rr} 329 & 1300 \\ 0 & 1319 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 531 & 20 \\ 1300 & 1187 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 1080 & 971 \end{array}\right),\left(\begin{array}{rr} 424 & 1315 \\ 45 & 14 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$1622016000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 6336.d
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 66.c3, its twist by $24$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{6}) \) | \(\Z/10\Z\) | 2.2.24.1-726.1-b3 |
$4$ | 4.0.2112.2 | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{6}, \sqrt{22})\) | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$8$ | 8.4.21159411204096.1 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.4857532416.6 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.2.42493693575168.4 | \(\Z/6\Z\) | Not in database |
$8$ | 8.0.40144896.1 | \(\Z/20\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | Not in database |
$16$ | 16.0.23595621172490797056.3 | \(\Z/2\Z \oplus \Z/20\Z\) | Not in database |
$16$ | deg 16 | \(\Z/30\Z\) | Not in database |
$20$ | 20.0.88908942498662728823242752000000000000000.2 | \(\Z/10\Z\) | 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 | add | ord | ord | nonsplit | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | - | 3 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-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.