Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-56x+3136\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-56xz^2+3136z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-72603x+146530998\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{6}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(6, 52\right)\) |
$\hat{h}(P)$ | ≈ | $0.52067737922981562558792047072$ |
Torsion generators
\( \left(28, 140\right) \)
Integral points
\( \left(-16, 8\right) \), \( \left(-14, 42\right) \), \( \left(-14, -28\right) \), \( \left(-12, 52\right) \), \( \left(-12, -40\right) \), \( \left(0, 56\right) \), \( \left(0, -56\right) \), \( \left(6, 52\right) \), \( \left(6, -58\right) \), \( \left(16, 72\right) \), \( \left(16, -88\right) \), \( \left(28, 140\right) \), \( \left(28, -168\right) \), \( \left(56, 392\right) \), \( \left(56, -448\right) \), \( \left(84, 728\right) \), \( \left(84, -812\right) \), \( \left(160, 1944\right) \), \( \left(160, -2104\right) \), \( \left(336, 5992\right) \), \( \left(336, -6328\right) \), \( \left(1666, 67172\right) \), \( \left(1666, -68838\right) \)
Invariants
Conductor: | \( 770 \) | = | $2 \cdot 5 \cdot 7 \cdot 11$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-4249907200 $ | = | $-1 \cdot 2^{12} \cdot 5^{2} \cdot 7^{3} \cdot 11^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{19443408769}{4249907200} \) | = | $-1 \cdot 2^{-12} \cdot 5^{-2} \cdot 7^{-3} \cdot 11^{-2} \cdot 2689^{3}$ | 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: | $0.52678270785344891664898630533\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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||
Stable Faltings height: | $0.52678270785344891664898630533\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.52067737922981562558792047072\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.1288970582858647608722881731\dots$ | 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: | $ 144 $ = $ ( 2^{2} \cdot 3 )\cdot2\cdot3\cdot2 $ | 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: | $6$ | 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) $ ≈ $ 2.3511646469141299204550955531 $ | 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
$\displaystyle 2.351164647 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.128897 \cdot 0.520677 \cdot 144}{6^2} \approx 2.351164647$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 576 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$11$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
$3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 924 = 2^{2} \cdot 3 \cdot 7 \cdot 11 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 874 & 915 \end{array}\right),\left(\begin{array}{rr} 913 & 12 \\ 912 & 13 \end{array}\right),\left(\begin{array}{rr} 317 & 2 \\ 366 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 673 & 12 \\ 342 & 73 \end{array}\right),\left(\begin{array}{rr} 543 & 850 \\ 98 & 401 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 142 & 3 \\ 633 & 916 \end{array}\right)$.
The torsion field $K:=\Q(E[924])$ is a degree-$1277337600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/924\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 770f
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | 4.2.13552.1 | \(\Z/12\Z\) | Not in database |
$6$ | 6.0.247066875.1 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.8999178496.3 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | 8.0.142355290000.8 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$9$ | 9.3.1356157919765880000.1 | \(\Z/18\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$18$ | 18.0.630833356046895622348746175819200000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | split | ord | nonsplit | split | nonsplit | ord | ss | ord | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 2 | 3 | 1 | 2 | 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 | 0 | 0 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.