Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+4207x+38657\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+4207xz^2+38657z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+67317x+2541382\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(15, 316\right) \) | $0.57300263832393287375524175373$ | $\infty$ |
| \( \left(-9, 4\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([15:316:1]\) | $0.57300263832393287375524175373$ | $\infty$ |
| \([-9:4:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(59, 2592\right) \) | $0.57300263832393287375524175373$ | $\infty$ |
| \( \left(-37, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-9, 4\right) \), \( \left(15, 316\right) \), \( \left(15, -332\right) \), \( \left(27, 400\right) \), \( \left(27, -428\right) \), \( \left(115, 1368\right) \), \( \left(115, -1484\right) \), \( \left(177, 2422\right) \), \( \left(177, -2600\right) \), \( \left(3741, 226954\right) \), \( \left(3741, -230696\right) \)
\([-9:4:1]\), \([15:316:1]\), \([15:-332:1]\), \([27:400:1]\), \([27:-428:1]\), \([115:1368:1]\), \([115:-1484:1]\), \([177:2422:1]\), \([177:-2600:1]\), \([3741:226954:1]\), \([3741:-230696:1]\)
\( \left(-37, 0\right) \), \((59,\pm 2592)\), \((107,\pm 3312)\), \((459,\pm 11408)\), \((707,\pm 20088)\), \((14963,\pm 1830600)\)
Invariants
| Conductor: | $N$ | = | \( 2790 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 31$ |
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| Minimal Discriminant: | $\Delta$ | = | $-5447624544000$ | = | $-1 \cdot 2^{8} \cdot 3^{11} \cdot 5^{3} \cdot 31^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{11298232190519}{7472736000} \) | = | $2^{-8} \cdot 3^{-5} \cdot 5^{-3} \cdot 19^{3} \cdot 31^{-2} \cdot 1181^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1326423388876128900009321958$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.58333619455355804430330957734$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9767197520246752$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.619142851316408$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.57300263832393287375524175373$ |
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| Real period: | $\Omega$ | ≈ | $0.47803969671613863877168310049$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{3}\cdot2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.3826881190707223986599438385 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.382688119 \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 0.478040 \cdot 0.573003 \cdot 64}{2^2} \\ & \approx 4.382688119\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7680 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $3$ | $4$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $31$ | $2$ | $I_{2}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 374 & 1 \\ 743 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1801 & 4 \\ 1742 & 9 \end{array}\right),\left(\begin{array}{rr} 622 & 1 \\ 619 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1857 & 4 \\ 1856 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 469 & 1396 \\ 464 & 1395 \end{array}\right)$.
The torsion field $K:=\Q(E[1860])$ is a degree-$164560896000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1860\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 45 = 3^{2} \cdot 5 \) |
| $3$ | additive | $8$ | \( 62 = 2 \cdot 31 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 558 = 2 \cdot 3^{2} \cdot 31 \) |
| $31$ | nonsplit multiplicative | $32$ | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 2790.p
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 930.h2, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.230640.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.175142250000.9 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.11968832160000.12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.2617583593392.2 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | nonsplit | ord | ord | ord | ord | ord | ord | ord | nonsplit | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 9 | - | 1 | 3 | 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 |
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.