Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+x^2-1296413083x-18035831105756\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+x^2z-1296413083xz^2-18035831105756z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1680151356000x-841459574253870000\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 88725 \) | = | $3 \cdot 5^{2} \cdot 7 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1071740411989715572969921875 $ | = | $-1 \cdot 3^{5} \cdot 5^{8} \cdot 7^{12} \cdot 13^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{756218111874334720}{3363432789843} \) | = | $-1 \cdot 2^{15} \cdot 3^{-5} \cdot 5 \cdot 7^{-12} \cdot 13 \cdot 73^{3} \cdot 97^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $4.0389720384851653386928998356\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.2560471918880739315900686524\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.125538162537704\dots$ | |||
Szpiro ratio: | $6.545064241598342\dots$ |
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: | $0.012576741536903090691338228952\dots$ | 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: | $ 180 $ = $ 5\cdot1\cdot( 2^{2} \cdot 3 )\cdot3 $ | 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) $ ≈ $ 2.2638134766425563244408812113 $ | 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
$\displaystyle 2.263813477 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.012577 \cdot 1.000000 \cdot 180}{1^2} \approx 2.263813477$
Modular invariants
Modular form 88725.2.a.bo
For more coefficients, see the Downloads section to the right.
Modular degree: | 50544000 | 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 $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
$5$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
$7$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
$13$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 |
---|---|---|
$3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 6.16.0-6.b.1.1, level \( 6 = 2 \cdot 3 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 2 & 3 \\ 1 & 4 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 3 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 2 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[6])$ is a degree-$18$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | good | $2$ | \( 12675 = 3 \cdot 5^{2} \cdot 13^{2} \) |
$3$ | split multiplicative | $4$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
$5$ | additive | $14$ | \( 1183 = 7 \cdot 13^{2} \) |
$7$ | split multiplicative | $8$ | \( 12675 = 3 \cdot 5^{2} \cdot 13^{2} \) |
$13$ | additive | $74$ | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 88725cf
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 88725o2, its twist by $65$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | not in database |
$3$ | 3.1.50700.1 | \(\Z/2\Z\) | not in database |
$3$ | 3.1.1026675.4 | \(\Z/3\Z\) | not in database |
$6$ | 6.0.7711470000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$6$ | 6.0.3162184666875.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
$9$ | 9.1.207778300343289000000.2 | \(\Z/6\Z\) | not in database |
$12$ | deg 12 | \(\Z/4\Z\) | not in database |
$18$ | 18.0.48629035529561149886617176061095733131591796875.2 | \(\Z/9\Z\) | not in database |
$18$ | 18.0.129515466280638029325742012563000000000000.1 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 | 13 |
---|---|---|---|---|---|
Reduction type | ss | split | add | split | add |
$\lambda$-invariant(s) | 4,3 | 3 | - | 1 | - |
$\mu$-invariant(s) | 0,0 | 1 | - | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $0$.