Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-24161x+1442310\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-24161xz^2+1442310z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-31312683x+67386353382\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(73, 226)$ | $0.13158555065930463261350455204$ | $\infty$ |
| $(367/4, -367/8)$ | $0$ | $2$ |
Integral points
\( \left(-179, 289\right) \), \( \left(-179, -110\right) \), \( \left(-77, 1726\right) \), \( \left(-77, -1649\right) \), \( \left(31, 835\right) \), \( \left(31, -866\right) \), \( \left(73, 226\right) \), \( \left(73, -299\right) \), \( \left(85, 25\right) \), \( \left(85, -110\right) \), \( \left(94, 16\right) \), \( \left(94, -110\right) \), \( \left(598, 13876\right) \), \( \left(598, -14474\right) \)
Invariants
| Conductor: | $N$ | = | \( 1365 \) | = | $3 \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $1481059636875$ | = | $3^{12} \cdot 5^{4} \cdot 7^{3} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{1559802282754777489}{1481059636875} \) | = | $3^{-12} \cdot 5^{-4} \cdot 7^{-3} \cdot 13^{-1} \cdot 23^{3} \cdot 50423^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2571869760059591717078545328$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2571869760059591717078545328$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9782957263783809$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.802966437034783$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.13158555065930463261350455204$ |
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| Real period: | $\Omega$ | ≈ | $0.84515383544607554352766770932$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ ( 2^{2} \cdot 3 )\cdot2\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.0017805909219133249115118347 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.001780591 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.845154 \cdot 0.131586 \cdot 72}{2^2} \\ & \approx 2.001780591\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3456 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 10913 & 8 \\ 10912 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3641 & 8 \\ 3644 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 8737 & 8 \\ 2188 & 33 \end{array}\right),\left(\begin{array}{rr} 848 & 3 \\ 10085 & 2 \end{array}\right),\left(\begin{array}{rr} 4099 & 4098 \\ 1378 & 6835 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 10914 & 10915 \end{array}\right),\left(\begin{array}{rr} 9364 & 1 \\ 4703 & 6 \end{array}\right),\left(\begin{array}{rr} 1368 & 6833 \\ 1393 & 1440 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$38954430627840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 91 = 7 \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 65 = 5 \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 273 = 3 \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 195 = 3 \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 105 = 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 1365d
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{91}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.37215879301758976.8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.39039316875.1 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | ord | split | nonsplit | split | ord | nonsplit | ord | ss | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 2 | 1 | 4 | 1 | 1 | 1 | 3,1 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.