Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2+2034x+60264\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z+2034xz^2+60264z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+32541x+3889438\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{93}{4}, \frac{93}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 306 \) | = | $2 \cdot 3^{2} \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-2133912053412 $ | = | $-1 \cdot 2^{2} \cdot 3^{22} \cdot 17 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1276229915423}{2927177028} \) | = | $2^{-2} \cdot 3^{-16} \cdot 17^{-1} \cdot 10847^{3}$ | 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: | $1.0498108396955544452268336144\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.50050469536149959952921099594\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0301007930420596\dots$ | |||
Szpiro ratio: | $6.211653920033552\dots$ |
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: | $0.57358701844345644344525798464\dots$ | 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\cdot2^{2}\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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 1.1471740368869128868905159693 $ | 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
$\displaystyle 1.147174037 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.573587 \cdot 1.000000 \cdot 8}{2^2} \approx 1.147174037$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 512 | 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 $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
$3$ | $4$ | $I_{16}^{*}$ | additive | -1 | 2 | 22 | 16 |
$17$ | $1$ | $I_{1}$ | nonsplit 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 | 16.48.0.132 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 816 = 2^{4} \cdot 3 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 812 & 813 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 271 & 0 \\ 0 & 815 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 718 & 803 \end{array}\right),\left(\begin{array}{rr} 13 & 288 \\ 60 & 121 \end{array}\right),\left(\begin{array}{rr} 568 & 273 \\ 543 & 10 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 801 & 16 \\ 800 & 17 \end{array}\right),\left(\begin{array}{rr} 178 & 483 \\ 399 & 430 \end{array}\right)$.
The torsion field $K:=\Q(E[816])$ is a degree-$481296384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/816\Z)$.
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 |
---|---|---|---|
$2$ | nonsplit multiplicative | $4$ | \( 153 = 3^{2} \cdot 17 \) |
$3$ | additive | $8$ | \( 34 = 2 \cdot 17 \) |
$17$ | nonsplit multiplicative | $18$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 306c
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 102b4, its twist by $-3$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$2$ | \(\Q(\sqrt{51}) \) | \(\Z/4\Z\) | not in database |
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | 2.0.3.1-3468.1-b2 |
$4$ | \(\Q(\sqrt{-3}, \sqrt{-17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{-2}, \sqrt{-3})\) | \(\Z/8\Z\) | not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{34})\) | \(\Z/8\Z\) | not in database |
$8$ | 8.0.500516630784.12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.4.128132257480704.36 | \(\Z/8\Z\) | not in database |
$8$ | 8.0.443364212736.7 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.0.98184462336.8 | \(\Z/16\Z\) | not in database |
$8$ | 8.2.236727913392.4 | \(\Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 17 |
---|---|---|---|
Reduction type | nonsplit | add | nonsplit |
$\lambda$-invariant(s) | 4 | - | 0 |
$\mu$-invariant(s) | 1 | - | 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 $0$.