Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-2475x-3003750\) | (homogenize, simplify) |
\(y^2z=x^3-2475xz^2-3003750z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-2475x-3003750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(150, 0\right) \)
Integral points
\( \left(150, 0\right) \)
Invariants
Conductor: | \( 61200 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-3896755776000000 $ | = | $-1 \cdot 2^{12} \cdot 3^{6} \cdot 5^{6} \cdot 17^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{35937}{83521} \) | = | $-1 \cdot 3^{3} \cdot 11^{3} \cdot 17^{-4}$ | 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: | $1.6705361601697068614578014023\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.37663612094134348095743300423\dots$ | 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: | $0.19972713735860544180302712530\dots$ | 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: | $ 64 $ = $ 2\cdot2\cdot2^{2}\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: | $2$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 3.1956341977376870688484340048 $ | 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
$\displaystyle 3.195634198 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.199727 \cdot 1.000000 \cdot 64}{2^2} \approx 3.195634198$
Modular invariants
Modular form 61200.2.a.gy
For more coefficients, see the Downloads section to the right.
Modular degree: | 262144 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{4}^{*}$ | Additive | -1 | 4 | 12 | 0 |
$3$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$5$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$17$ | $4$ | $I_{4}$ | 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 | 16.48.0.76 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 16320 = 2^{6} \cdot 3 \cdot 5 \cdot 17 \), index $1536$, genus $53$, and generators
$\left(\begin{array}{rr} 1 & 64 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 16257 & 64 \\ 16256 & 65 \end{array}\right),\left(\begin{array}{rr} 914 & 225 \\ 7965 & 3854 \end{array}\right),\left(\begin{array}{rr} 3263 & 0 \\ 0 & 16319 \end{array}\right),\left(\begin{array}{rr} 57 & 16 \\ 9968 & 14537 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 64 & 1 \end{array}\right),\left(\begin{array}{rr} 11071 & 4830 \\ 3150 & 15511 \end{array}\right),\left(\begin{array}{rr} 9 & 124 \\ 8372 & 8361 \end{array}\right),\left(\begin{array}{rr} 12481 & 6540 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10879 & 0 \\ 0 & 16319 \end{array}\right)$.
The torsion field $K:=\Q(E[16320])$ is a degree-$7392712458240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/16320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 61200.gy
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 17.a3, its twist by $60$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{15}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-15}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.3317760000.4 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.4.1082432160000.7 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.17318914560000.125 | \(\Z/8\Z\) | Not in database |
$8$ | 8.2.29225668320000.1 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | 17 |
---|---|---|---|---|
Reduction type | add | add | add | split |
$\lambda$-invariant(s) | - | - | - | 1 |
$\mu$-invariant(s) | - | - | - | 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 $0$.