Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-454x+5812\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-454xz^2+5812z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7267x+364702\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(23, 73)$ | $0.19763543254554395156660112286$ | $\infty$ |
Integral points
\( \left(-3, 86\right) \), \( \left(-3, -83\right) \), \( \left(13, 38\right) \), \( \left(13, -51\right) \), \( \left(23, 73\right) \), \( \left(23, -96\right) \)
Invariants
| Conductor: | $N$ | = | \( 338 \) | = | $2 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-8031810176$ | = | $-1 \cdot 2^{7} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{2146689}{1664} \) | = | $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 13^{-1} \cdot 43^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.59931413030649628067007441908$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.68316054842427208735666930170$ |
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| $abc$ quality: | $Q$ | ≈ | $0.96783604338842$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.292599722814764$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.19763543254554395156660112286$ |
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| Real period: | $\Omega$ | ≈ | $1.2055736043537517075370810582$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $0.95305624304777675025345814826 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 0.953056243 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.205574 \cdot 0.197635 \cdot 4}{1^2} \\ & \approx 0.953056243\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 336 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 |
|---|---|---|
| $7$ | 7B.6.1 | 7.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 728 = 2^{3} \cdot 7 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 547 & 378 \\ 0 & 339 \end{array}\right),\left(\begin{array}{rr} 608 & 721 \\ 287 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 357 & 722 \end{array}\right),\left(\begin{array}{rr} 715 & 14 \\ 714 & 15 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 175 & 722 \end{array}\right)$.
The torsion field $K:=\Q(E[728])$ is a degree-$845365248$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/728\Z)$.
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 | $4$ | \( 169 = 13^{2} \) |
| $7$ | good | $2$ | \( 169 = 13^{2} \) |
| $13$ | additive | $98$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 338f
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 26b1, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/7\Z\) | 2.2.13.1-52.1-b2 |
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.140608.1 | \(\Z/14\Z\) | not in database |
| $8$ | 8.2.168899700528.2 | \(\Z/3\Z\) | not in database |
| $12$ | 12.2.8421963387109376.8 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.1265319018496.1 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $16$ | deg 16 | \(\Z/21\Z\) | 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 | nonsplit | ss | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 2 | 1,1 | 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 |
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.