Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+33744x-1075372\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+33744xz^2-1075372z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2733237x-792145926\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(316, 6422)$ | $0.68200879828789502756961116003$ | $\infty$ |
| $(31, 38)$ | $3.7829192117137131220035736405$ | $\infty$ |
Integral points
\((31,\pm 38)\), \((43,\pm 676)\), \((316,\pm 6422)\)
Invariants
| Conductor: | $N$ | = | \( 51376 \) | = | $2^{4} \cdot 13^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-2969071076900864$ | = | $-1 \cdot 2^{17} \cdot 13^{7} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{214921799}{150176} \) | = | $2^{-5} \cdot 13^{-1} \cdot 19^{-2} \cdot 599^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6574441033057873871828260819$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.31817775598492629026114976034$ |
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| $abc$ quality: | $Q$ | ≈ | $0.860970828213973$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.954414869198532$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5191480733166642817147091216$ |
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| Real period: | $\Omega$ | ≈ | $0.25466719022417489080844843534$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.264709785443177873019677159 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.264709785 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.254667 \cdot 2.519148 \cdot 16}{1^2} \\ & \approx 10.264709785\end{aligned}$$
Modular invariants
Modular form 51376.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 161280 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{9}^{*}$ | additive | -1 | 4 | 17 | 5 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $19$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 104 = 2^{3} \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 53 & 2 \\ 53 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 103 & 2 \\ 102 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 103 & 0 \end{array}\right),\left(\begin{array}{rr} 79 & 2 \\ 79 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[104])$ is a degree-$20127744$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/104\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 169 = 13^{2} \) |
| $13$ | additive | $98$ | \( 304 = 2^{4} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 2704 = 2^{4} \cdot 13^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 51376.s consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 494.a1, its twist by $-52$.
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 |
|---|---|---|---|
| $3$ | 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 |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | add | ord | ord | ord | ss | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | 2 | 2 | 2,2 | - | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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.