Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+6971x-20944\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+6971xz^2-20944z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+9035037x-1004256738\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{75}{4}, \frac{2651}{8}\right) \) | $4.9369404894194568248580123243$ | $\infty$ |
| \( \left(3, -2\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([150:2651:8]\) | $4.9369404894194568248580123243$ | $\infty$ |
| \([3:-2:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(678, 73710\right) \) | $4.9369404894194568248580123243$ | $\infty$ |
| \( \left(111, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(3, -2\right) \)
\([3:-2:1]\)
\( \left(111, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 11130 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 53$ |
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| Minimal Discriminant: | $\Delta$ | = | $-21884919000000$ | = | $-1 \cdot 2^{6} \cdot 3 \cdot 5^{6} \cdot 7^{2} \cdot 53^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{37471278716561591}{21884919000000} \) | = | $2^{-6} \cdot 3^{-1} \cdot 5^{-6} \cdot 7^{-2} \cdot 11^{3} \cdot 29^{3} \cdot 53^{-3} \cdot 1049^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2491543949311848922584416498$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2491543949311848922584416498$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9843933697975319$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.0958157221663525$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.9369404894194568248580123243$ |
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| Real period: | $\Omega$ | ≈ | $0.40042812529763473478533577263$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9537796497684408162313861728 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.953779650 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.400428 \cdot 4.936940 \cdot 8}{2^2} \\ & \approx 3.953779650\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 27648 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $53$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $3$ | 3B.1.2 | 3.8.0.2 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 44520 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 53 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 44509 & 12 \\ 44508 & 13 \end{array}\right),\left(\begin{array}{rr} 10090 & 3 \\ 8373 & 44512 \end{array}\right),\left(\begin{array}{rr} 9279 & 1858 \\ 42686 & 35261 \end{array}\right),\left(\begin{array}{rr} 7430 & 3 \\ 29653 & 44512 \end{array}\right),\left(\begin{array}{rr} 25441 & 12 \\ 19086 & 73 \end{array}\right),\left(\begin{array}{rr} 35617 & 12 \\ 35622 & 73 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 44470 & 44511 \end{array}\right),\left(\begin{array}{rr} 22261 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[44520])$ is a degree-$5751343436267520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/44520\Z)$.
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 | $4$ | \( 159 = 3 \cdot 53 \) |
| $3$ | split multiplicative | $4$ | \( 7 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 2226 = 2 \cdot 3 \cdot 7 \cdot 53 \) |
| $7$ | split multiplicative | $8$ | \( 1590 = 2 \cdot 3 \cdot 5 \cdot 53 \) |
| $53$ | nonsplit multiplicative | $54$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 11130.n
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-159}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/6\Z\) | not in database |
| $3$ | 3.1.11907.1 | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.12465600.6 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{53})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.425329947.3 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.63321846519519.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.1749257125384239939612155777985942573897627000000000000.3 | \(\Z/18\Z\) | 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 | 53 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit | split | ss | ord | ord | ord | ss | ss | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 3 | 6 | 1 | 2 | 1,1 | 1 | 1 | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 1 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.